Matrices | ISC Class 12 Maths Notes

This note covers matrix notation and order, types of matrices, equality, addition and subtraction, scalar multiplication, matrix multiplication, transpose, symmetric and skew-symmetric matrices, and invertible matrices with the proof of uniqueness of an inverse.

What is a matrix, and how do we read its order and entries?

Definition: A matrix is an ordered rectangular arrangement of numbers or functions. Each number or function in the arrangement is an element, also called an entry. Here, entries are real numbers or functions taking real values.

A row runs horizontally and a column runs vertically. Capital letters such as A name matrices. Position matters: moving an entry to a different row or column can change the matrix, even when the collection of numbers remains the same.

How does the notation identify a position?

Write $A=[a_{ij}]_{m\times n}$, where m is the number of rows, n is the number of columns, i is the row number, and j is the column number. The symbol $a_{ij}$ denotes the entry in row i and column j.

The order is m × n, read as “m by n”. Both counts are positive integers. The row index i runs from 1 to m; the column index j runs from 1 to n. There are mn entries, where mn means m multiplied by n.

Read the row count first. A matrix with three rows and two columns has order 3 × 2. Reversing these counts gives a different shape. Knowing the total number of entries alone does not specify how they are arranged.

Worked example 1. Three factories, labelled I, II and III, have respectively 30, 25 and 27 men workers, and 25, 31 and 26 women workers. Represent the information with factories as rows and men and women as the first and second columns.

Answer: $A=\begin{bmatrix}30&25\\25&31\\27&26\end{bmatrix}$. Its order is 3 × 2, and it contains six entries. The entry $a_{32}=26$ records the women workers in factory III.

The row and column labels give the numbers their meaning. Once their order has been chosen, keep that order throughout an operation. Reading an entry requires both its position and the meaning assigned to the corresponding row and column.

How are the different types of matrices identified?

Matrix types describe shape or restrictions on entries. A row matrix has one row; a column matrix has one column. A square matrix has the same number of rows and columns. Its order may be stated as n instead of n × n.

In a square matrix, the main diagonal consists of entries with equal row and column indices, from the top left to the bottom right. An off-diagonal entry is one whose row and column indices differ.

How do these descriptions overlap?

The symbol $I_n$ denotes the identity matrix of order n; write I when the order is clear. The symbol O denotes a zero matrix of the required order. For order 2, $I_2=\begin{bmatrix}1&0\\0&1\end{bmatrix}$.

Every identity matrix is scalar, and every scalar matrix is diagonal. A scalar matrix is an identity matrix when its common diagonal entry is 1. A diagonal matrix need not have equal diagonal entries: $\begin{bmatrix}-1&0\\0&2\end{bmatrix}$ is diagonal but not scalar.

The word triangular specifies which side of the diagonal must contain zeros. It does not require the diagonal entries to be equal. A diagonal matrix satisfies both triangular conditions because all entries on both sides of the diagonal are zero.

Note: A matrix can satisfy more than one definition. Identify its shape first, then examine the diagonal and off-diagonal entries. Do not treat the names of matrix types as mutually exclusive categories.

How do we construct matrices and use equality of matrices?

A formula for the general entry tells us how to fill each position. Substitute its row and column numbers into the formula. Keep the indices in their stated roles: exchanging i and j changes the substitution unless the formula happens to give the same result.

How is an entry formula applied?

Worked example 2. Construct a 3 × 2 matrix A with $a_{ij}=\frac12|i-3j|$. Here i is the row number, j is the column number, and vertical bars around a real expression mean its absolute value, the non-negative magnitude of that expression.

Answer: In the first row, $a_{11}=\frac12|1-3|=1$ and $a_{12}=\frac12|1-6|=5/2$. The second row gives $a_{21}=1/2$ and $a_{22}=2$; the third gives $a_{31}=0$ and $a_{32}=3/2$.

Thus $A=\begin{bmatrix}1&5/2\\1/2&2\\0&3/2\end{bmatrix}$. Each row uses a fixed i, while j changes across that row.

What must be true for two matrices to be equal?

Equality of matrices requires the same order and equality of every pair of corresponding entries. If B has entries $b_{ij}$, then A = B requires $a_{ij}=b_{ij}$ at every position. Having equal numbers of entries is not sufficient.

Worked example 3. Find the unknown real numbers a, b, c and d if $\begin{bmatrix}2a+b&a-2b\\5c-d&4c+3d\end{bmatrix}=\begin{bmatrix}4&-3\\11&24\end{bmatrix}$.

Answer: Corresponding entries give 2a + b = 4, a − 2b = −3, 5c − d = 11 and 4c + 3d = 24. Solving the first pair gives a = 1 and b = 2. Solving the second gives c = 3 and d = 4.

Check the solution in the original positions: the first row becomes 4, −3 and the second becomes 11, 24. Equality is a statement about the complete arrangement, so every corresponding entry must agree, including entries that contain no unknowns.

How do matrix addition and subtraction work?

Two matrices can be added or subtracted when they have the same order. Combine corresponding entries. The resulting matrix keeps that order. A shared number of rows alone, or a shared number of columns alone, does not meet the condition.

For matrices A and B with entries $a_{ij}$ and $b_{ij}$, the sum C = A + B has entries $c_{ij}=a_{ij}+b_{ij}$. Here $c_{ij}$ names the entry of C in row i and column j.

The negative of A, written −A, is obtained by changing the sign of each entry. The difference A − B is A + (−B). Its entry at each position is the corresponding entry of A minus that of B.

Property: addition is commutative and associative

Commutative means that changing the order of addition leaves the result unchanged: A + B = B + A. This follows because ordinary real-number addition gives the same answer at each corresponding position in either order.

Associative means that changing the grouping leaves the result unchanged. For A, B and C of the same order, (A + B) + C = A + (B + C). The brackets specify which sum is evaluated first.

What are the additive identity and additive inverse?

The additive identity is the zero matrix of the same order: A + O = O + A = A. Adding its zero entries leaves every entry of A unchanged. Its dimensions must still agree with those of A.

The additive inverse of A is −A, since A + (−A) = (−A) + A = O. This is an inverse for addition. The inverse associated with matrix multiplication has a different definition and need not exist.

Note: Subtraction applies to every entry of the second matrix. A negative entry being subtracted contributes a positive amount. Write the corresponding scalar calculations before combining them if the signs are difficult to track.

How do scalar multiplication and matrix equations work?

A scalar is a real number used to multiply a matrix. For a scalar k, the matrix kA is formed by multiplying every entry of A by k. Its order remains unchanged. In particular, −A = (−1)A.

Property: scalar multiplication distributes over addition

For A and B of the same order, k(A + B) = kA + kB. If k and l are real scalars, then (k + l)A = kA + lA. Each identity follows by applying ordinary numerical multiplication and addition to corresponding entries.

Worked example 4. Given $A=\begin{bmatrix}1&2&3\\2&3&1\end{bmatrix}$ and $B=\begin{bmatrix}3&-1&3\\-1&0&2\end{bmatrix}$, find 2A − B.

Answer: First, $2A=\begin{bmatrix}2&4&6\\4&6&2\end{bmatrix}$. Subtract corresponding entries to obtain $2A-B=\begin{bmatrix}-1&5&3\\5&6&0\end{bmatrix}$. In particular, the first-row second entry is 4 − (−1) = 5, and the second-row first entry is 4 − (−1) = 5.

How can two unknown matrices be separated?

Matrix equations involving sums can be combined using the laws of addition. If the sum and difference of two unknown matrices are given, adding the equations removes one unknown. Subtracting the equations removes the other. Multiplication by one half then gives each matrix.

Worked example 5. Find the unknown matrices X and Y if $X+Y=\begin{bmatrix}5&2\\0&9\end{bmatrix}$ and $X-Y=\begin{bmatrix}3&6\\0&-1\end{bmatrix}$.

Answer: Adding gives $2X=\begin{bmatrix}8&8\\0&8\end{bmatrix}$, hence $X=\begin{bmatrix}4&4\\0&4\end{bmatrix}$. Subtracting the second equation from the first gives $2Y=\begin{bmatrix}2&-4\\0&10\end{bmatrix}$, hence $Y=\begin{bmatrix}1&-2\\0&5\end{bmatrix}$.

Substitution checks both original equations. The sum of the displayed answers gives the required sum matrix, while their difference gives the required difference matrix. These steps use scalar multiplication and addition, so all entries remain in their corresponding positions.

When is matrix multiplication possible, and how is it calculated?

The product AB is defined when the number of columns of A equals the number of rows of B. These are the compatible dimensions. If A has order m × n and B has order n × p, then AB has order m × p.

Here p is the number of columns of B. To find an entry in row i and column j of AB, multiply the entries of row i of A by the corresponding entries of column j of B, then add those products.

With C = AB and $c_{ij}$ its row-i, column-j entry, the rule is $c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{in}b_{nj}$. The dots mean that the same pattern continues through the shared dimension n.

How is a complete product assembled?

1. Record both orders and check that the inner dimensions agree.
2. Use the outer dimensions to determine the order of the answer.
3. For each answer position, select its row from the first matrix and its column from the second.
4. Multiply the paired entries, add the products, and place the sum in that position.

Worked example 6. Find AB for $A=\begin{bmatrix}6&9\\2&3\end{bmatrix}$ and $B=\begin{bmatrix}2&6&0\\7&9&8\end{bmatrix}$.

Answer: The orders 2 × 2 and 2 × 3 are compatible, giving a 2 × 3 product. The first row is 6 × 2 + 9 × 7 = 75, 6 × 6 + 9 × 9 = 117, and 6 × 0 + 9 × 8 = 72.

The second row is 2 × 2 + 3 × 7 = 25, 2 × 6 + 3 × 9 = 39, and 2 × 0 + 3 × 8 = 24. Thus $AB=\begin{bmatrix}75&117&72\\25&39&24\end{bmatrix}$.

For these same matrices, BA is not defined: B has three columns but A has two rows. Reversing the order therefore requires a fresh compatibility check. A defined product in one direction does not guarantee a defined product in the other.

Which multiplication rules hold, and which numerical rules fail?

Property: multiplication is associative and distributive

For matrices A, B and C, associativity gives (AB)C = A(BC), whenever both sides are defined. This changes the grouping without changing the sequence of factors. It does not permit exchanging the positions of A and B.

Distributivity gives A(B + C) = AB + AC and (A + B)C = AC + BC, whenever both sides are defined. The unchanged factor stays on its original side. For a square matrix A, the identity matrix of the same order satisfies IA = AI = A.

Why can the order of multiplication matter?

Matrix multiplication is not commutative: even if both products are defined and have the same order, they need not be equal. Matrices that satisfy AB = BA are said to commute. This equality must be justified for the matrices concerned.

Worked example 7. Compare AB and BA for $A=\begin{bmatrix}1&0\\0&-1\end{bmatrix}$ and $B=\begin{bmatrix}0&1\\1&0\end{bmatrix}$.

Answer: Row-by-column multiplication gives $AB=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$, whereas $BA=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$. Their off-diagonal entries differ, so AB ≠ BA even though both are square matrices of order 2.

This does not mean that AB ≠ BA for every pair of matrices for which both products are defined. Diagonal matrices of the same order commute. Non-commutativity describes the absence of a general equality, not a requirement that every pair give unequal products.

Can non-zero matrices have a zero product?

Worked example 8. Find AB for $A=\begin{bmatrix}0&-1\\0&2\end{bmatrix}$ and $B=\begin{bmatrix}3&5\\0&0\end{bmatrix}$.

Answer: The first row of the product is 0 × 3 + (−1) × 0 = 0 and 0 × 5 + (−1) × 0 = 0. The second row also contains two zeros. Thus $AB=\begin{bmatrix}0&0\\0&0\end{bmatrix}=O$, although neither factor is zero.

For real numbers, a zero product requires at least one zero factor. The example shows why that inference need not be true for matrices. Inspect the row-by-column products instead of applying the numerical rule without checking it.

What is a transpose, and how does it affect operations?

The transpose of A is the matrix obtained by interchanging its rows and columns. It is written A′, read as “A transpose”. The first row becomes the first column, the second row becomes the second column, and the pattern continues.

If A has order m × n, A′ has order n × m. The entry originally in row i and column j moves to row j and column i. Transposition moves entries; it does not change their numerical values or signs.

Property: transpose rules preserve sums but reverse products

Worked example 9. For $A=\begin{bmatrix}3&3&2\\4&2&0\end{bmatrix}$, find A′ and verify (A′)′ = A.

Answer: The first row becomes the first column and the second row becomes the second column, giving $A'=\begin{bmatrix}3&4\\3&2\\2&0\end{bmatrix}$. Its order is 3 × 2. Transposing again gives $(A')'=\begin{bmatrix}3&3&2\\4&2&0\end{bmatrix}=A$.

The product rule deserves a separate check. If A has order m × n and B has order n × p, then B′ has order p × n and A′ has order n × m. Their product B′A′ therefore has order p × m, matching (AB)′.

Note: The rule for a sum keeps the order of the terms, but the rule for a product reverses the factors. Copying the sum pattern directly into a product can give an incorrect result or an undefined multiplication.

How are symmetric and skew-symmetric matrices recognised and formed?

A square matrix A is symmetric if A′ = A. Corresponding entries across the main diagonal are equal. It is skew-symmetric if A′ = −A, so corresponding entries across the diagonal are negatives of one another.

Every diagonal entry of a real skew-symmetric matrix is zero. Indeed, a diagonal entry must equal its own negative. Adding the entry to both sides gives twice that entry equal to zero, which forces the entry itself to be zero.

Theorem: transpose sums and differences have special symmetry

For any real square matrix A, A + A′ is symmetric. Its transpose is A′ + (A′)′ = A′ + A = A + A′. This uses the transpose-of-a-sum rule, double transposition and commutativity of addition.

Similarly, A − A′ is skew-symmetric: its transpose is A′ − A = −(A − A′). The square-matrix condition matters because A and A′ must have the same order before these sums and differences are formed.

Theorem: every square matrix has a symmetric and skew-symmetric decomposition

A decomposition here means writing a matrix as a sum of two matrices. Define P = (A + A′)/2 and Q = (A − A′)/2, where division by 2 means scalar multiplication by one half. Then P′ = P, Q′ = −Q, and A = P + Q.

Worked example 10. Decompose $B=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}$ into a symmetric matrix P and a skew-symmetric matrix Q.

Answer: $B'=\begin{bmatrix}2&-1&1\\-2&3&-2\\-4&4&-3\end{bmatrix}$. Taking half the sum gives $P=\begin{bmatrix}2&-3/2&-3/2\\-3/2&3&1\\-3/2&1&-3\end{bmatrix}$. Taking half the difference gives $Q=\begin{bmatrix}0&-1/2&-5/2\\1/2&0&3\\5/2&-3&0\end{bmatrix}$.

The matching entries of P are equal, the matching off-diagonal entries of Q have opposite signs, and Q has zero diagonal. Adding P and Q returns B, establishing the required decomposition.

For symmetric matrices A and B of the same order, AB is symmetric if and only if they commute. “If and only if” means both implications hold. Since (AB)′ = B′A′ = BA, symmetry of AB is equivalent to AB = BA.

What is an invertible matrix, and why is its inverse unique if it exists?

Definition: A square matrix A is invertible if a square matrix B of the same order satisfies AB = BA = I. Then B is the inverse of A, written A⁻¹. The superscript −1 here denotes a matrix inverse, not reciprocals of individual entries.

The inverse is defined through multiplication on both sides. If B is the inverse of A, then A is the inverse of B. Being square is necessary for this definition, but it does not by itself guarantee an inverse.

How can a proposed inverse be verified?

Worked example 11. Verify that $B=\begin{bmatrix}2&-3\\-1&2\end{bmatrix}$ is the inverse of $A=\begin{bmatrix}2&3\\1&2\end{bmatrix}$.

Answer: $AB=\begin{bmatrix}4-3&-6+6\\2-2&-3+4\end{bmatrix}=I_2$. In the reverse order, $BA=\begin{bmatrix}4-3&6-6\\-2+2&-3+4\end{bmatrix}=I_2$. Both products equal the identity, so B = A⁻¹.

Theorem: uniqueness of the inverse, if it exists

Let A be square and suppose B and C are both inverses of A. Then AB = BA = I and AC = CA = I. To prove uniqueness, show that the two proposed inverses must be equal.

1. Begin with B = BI, because multiplication by the identity leaves B unchanged.
2. Replace I with AC to obtain B = B(AC), using the assumption that C is an inverse.
3. Use associativity to write B(AC) = (BA)C without changing the order of any factors.
4. Replace BA with I, giving B = IC = C. Thus the inverse, if it exists, is unique.

How do singularity and invertibility relate?

The determinant is a scalar associated with a square matrix, written det(A) or |A|. For $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, where a, b, c and d are its real entries, det(A) = ad − bc.

A square matrix is singular when its determinant is zero and non-singular when its determinant is non-zero. A square matrix is invertible if and only if it is non-singular. Vertical bars around a matrix denote its determinant, unlike absolute-value bars around a real expression.

For invertible matrices A and B of the same order, (AB)⁻¹ = B⁻¹A⁻¹. The factors reverse order. The statement assumes invertibility of both matrices; the uniqueness theorem likewise proves that two inverses cannot differ, rather than asserting that every square matrix has an inverse.

Glossary

• Matrix — An ordered rectangular array of numbers or functions arranged in rows and columns.
• Entry — An individual number or function occupying a specified row and column of a matrix.
• Order — The number of rows followed by the number of columns in a matrix.
• Main diagonal — The entries of a square matrix whose row and column indices are equal.
• Diagonal matrix — A square matrix whose entries outside the main diagonal are all zero.
• Scalar matrix — A diagonal matrix in which every diagonal entry has the same value.
• Identity matrix — A square matrix with diagonal entries one and all other entries zero.
• Zero matrix — A matrix in which every entry is zero, regardless of its order.
• Scalar multiplication — Multiplication of every entry of a matrix by the same real number.
• Transpose — The matrix obtained by interchanging the rows and columns of a given matrix.
• Symmetric matrix — A square matrix equal to its transpose, with matching entries across its diagonal.
• Skew-symmetric matrix — A square matrix whose transpose equals its negative and whose diagonal entries are zero.
• Inverse matrix — A matrix giving the identity when multiplied with the original square matrix in either order.
• Singular matrix — A square matrix whose determinant is zero and which therefore has no inverse.
• Commuting matrices — Matrices for which both product orders are defined and give the same matrix.

Common errors and misconceptions

• Misconception: Matrices are equal whenever they contain the same numbers. Correct: They must have the same order and equal entries at every corresponding position. Rearranging the same numbers can produce a different matrix.
• Misconception: Matrices of different orders can be added if they contain the same number of entries. Correct: Addition requires the same row count and the same column count, so corresponding positions can be paired.
• Misconception: Matrix multiplication means multiplying corresponding entries. Correct: Each product entry is a sum of paired products from a row of the first matrix and a column of the second.
• Misconception: If AB exists, BA exists and equals AB. Correct: BA needs its own compatibility check. Even when both products exist and have the same order, their entries need not agree.
• Misconception: A zero matrix product proves that one factor is zero. Correct: Two non-zero matrices can have a zero product. The numerical zero-product inference need not hold for matrices.
• Misconception: Transposing a product preserves the order of the factors. Correct: The rule is (AB)′ = B′A′. Both factors are transposed and their order is reversed.
• Misconception: A zero diagonal is sufficient to prove skew-symmetry. Correct: The off-diagonal entries must also satisfy the opposite-sign condition at every pair of positions across the main diagonal.
• Misconception: Every square matrix has an inverse obtained by taking entry-wise reciprocals. Correct: Invertibility requires a matrix whose products in both orders are the identity. The inverse, if it exists, is unique.

Exam-style questions with model answers

Q1. For $A=\begin{bmatrix}30&25\\25&31\\27&26\end{bmatrix}$, state its order and number of entries, and identify the entry in the third row and second column. [2 marks]

1. A has three rows and two columns, so its order is 3 × 2 and it contains six entries.
2. The entry in the third row and second column is 26; the row number is read before the column number.

Q2. Construct a 3 × 2 matrix A whose row-i, column-j entry is $a_{ij}=\frac12|i-3j|$, where the bars mean absolute value. [3 marks]

1. Use i = 1 for the first row and j = 1, 2 for its columns. The entries are half of |1 − 3| and |1 − 6|, giving 1 and 5/2.
2. For i = 2, the two entries are half of |2 − 3| and |2 − 6|, giving 1/2 and 2.
3. For i = 3, the entries are 0 and 3/2. Therefore $A=\begin{bmatrix}1&5/2\\1/2&2\\0&3/2\end{bmatrix}$, arranged in the required three rows and two columns.

Q3. Find the real numbers a, b, c and d if $\begin{bmatrix}2a+b&a-2b\\5c-d&4c+3d\end{bmatrix}=\begin{bmatrix}4&-3\\11&24\end{bmatrix}$. [4 marks]

1. The matrices have the same order. Equating corresponding entries in the first row gives 2a + b = 4 and a − 2b = −3.
2. From the second equation, a = 2b − 3. Substituting into the first gives 5b = 10, so b = 2 and a = 1.
3. The second row gives 5c − d = 11 and 4c + 3d = 24. Thus d = 5c − 11.
4. Substitution gives 19c = 57, hence c = 3 and d = 4. These values satisfy all four corresponding-entry equations.

Q4. Given $A=\begin{bmatrix}1&2&3\\2&3&1\end{bmatrix}$ and $B=\begin{bmatrix}3&-1&3\\-1&0&2\end{bmatrix}$, calculate 2A − B. [3 marks]

1. Both matrices have order 2 × 3, so the subtraction is defined. Multiplying every entry of A by 2 gives $2A=\begin{bmatrix}2&4&6\\4&6&2\end{bmatrix}$.
2. Subtract corresponding entries in the first row: 2 − 3 = −1, 4 − (−1) = 5, and 6 − 3 = 3.
3. The second row gives 4 − (−1) = 5, 6 − 0 = 6, and 2 − 2 = 0. Therefore $2A-B=\begin{bmatrix}-1&5&3\\5&6&0\end{bmatrix}$.

Q5. For $A=\begin{bmatrix}6&9\\2&3\end{bmatrix}$ and $B=\begin{bmatrix}2&6&0\\7&9&8\end{bmatrix}$, calculate AB and explain whether BA is defined. [4 marks]

1. A has order 2 × 2 and B has order 2 × 3. Their inner dimensions agree, so AB exists and has order 2 × 3.
2. Multiplying the first row of A by the three columns of B gives 75, 117 and 72 respectively.
3. The second row gives 25, 39 and 24. Thus $AB=\begin{bmatrix}75&117&72\\25&39&24\end{bmatrix}$.
4. BA is not defined because the three columns of B do not match the two rows of A. Compatibility must be checked in the requested order.

Q6. Let $A=\begin{bmatrix}1&0\\0&-1\end{bmatrix}$, $B=\begin{bmatrix}0&1\\1&0\end{bmatrix}$, $C=\begin{bmatrix}0&-1\\0&2\end{bmatrix}$ and $D=\begin{bmatrix}3&5\\0&0\end{bmatrix}$. Use AB, BA and CD to demonstrate two ways matrix multiplication differs from real-number multiplication. [5 marks]

1. All four matrices have order 2 × 2, so the requested products exist. Multiplying A by B gives $AB=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$.
2. Reversing the factors gives $BA=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$. Its off-diagonal entries differ from the corresponding entries of AB, so AB ≠ BA.
3. This demonstrates that matrix multiplication is not commutative. It does not assert that every pair of matrices has unequal products in the two orders.
4. For CD, the first row gives 0 × 3 + (−1) × 0 = 0 and 0 × 5 + (−1) × 0 = 0. The second row also gives zeros, so CD = O.
5. C and D each contain non-zero entries. Thus non-zero matrix factors can have a zero product, whereas for real numbers a zero product requires at least one zero factor.

Q7. Prove that any square matrix A with real entries can be written as the sum of a symmetric matrix and a skew-symmetric matrix. Use A′ to denote its transpose. [5 marks]

1. A is square, so A and A′ have the same order. Define P = (A + A′)/2, where division by 2 means multiplying every entry by one half.
2. Taking the transpose gives P′ = (A′ + (A′)′)/2 = (A′ + A)/2 = P. Therefore P is symmetric.
3. Define Q = (A − A′)/2. Its transpose is Q′ = (A′ − A)/2 = −Q, so Q is skew-symmetric.
4. Add the two expressions: P + Q = (A + A′ + A − A′)/2 = 2A/2 = A, because the transpose terms cancel under addition.
5. Hence A = P + Q is the required decomposition. The formulas establish both the symmetry conditions and recovery of the original matrix, for every real square matrix A.

Q8. Prove that the inverse of a square matrix A, if it exists, is unique. Use I for the identity matrix of the same order. [5 marks]

1. Suppose B and C are both inverses of A. By the definition of inverse, AB = BA = I and AC = CA = I.
2. Start with B = BI, since I is the multiplicative identity. Replace I with AC to obtain B = B(AC), using the inverse relation for C.
3. Apply associativity of matrix multiplication: B(AC) = (BA)C. This changes the grouping of the factors while preserving their order, so no commutativity assumption is needed.
4. Since BA = I, the expression becomes (BA)C = IC = C. Combining these equalities gives B = C.
5. Thus any two proposed inverses of A are equal, proving uniqueness if an inverse exists. The argument assumes existence and does not claim that every square matrix is invertible.

Key takeaways

• A matrix is an ordered arrangement: its order records rows before columns, and each entry has a fixed position.
• Equality requires the same order and equal corresponding entries; addition and subtraction also require matching orders.
• Scalar multiplication acts on every entry and preserves the matrix order, including when the scalar is negative.
• For AB to exist, the columns of A must match the rows of B; the outer dimensions determine the product order.
• Matrix multiplication is associative and distributive where defined, but it is not commutative in general.
• A product can be zero even when neither matrix factor is zero, so numerical zero-product reasoning cannot be transferred directly.
• Transposing swaps rows and columns, while transposing a product also reverses the order of its factors.
• A square matrix splits into half its transpose sum and half its transpose difference, giving symmetric and skew-symmetric parts.
• An inverse gives the identity in both multiplication orders and, if it exists, is unique.

Test yourself

How do row and column indices identify an entry?

The first index identifies the row and the second identifies the column. Their order determines the entry's position in the matrix.

What distinguishes a scalar matrix from a diagonal matrix?

A scalar matrix has equal diagonal entries as well as zero off-diagonal entries. A diagonal matrix does not require its diagonal entries to be equal.

What two conditions establish equality of matrices?

The matrices must have the same order, and every entry must equal the entry in the corresponding position of the other matrix.

If A has order 2 × 2 and B has order 2 × 3, which of AB and BA is defined?

AB is defined and has order 2 × 3. BA is not defined because three columns would need to match two rows.

Does non-commutativity mean that no matrices commute?

No. Some matrices commute, including diagonal matrices of the same order. The general claim that reversing factors preserves every product is false.

Why are the diagonal entries of a real skew-symmetric matrix zero?

Each diagonal entry equals its own negative. Twice that entry is therefore zero, which forces the entry to be zero.

For a square matrix A with transpose A′, what are its symmetric and skew-symmetric parts?

The symmetric part is (A + A′)/2 and the skew-symmetric part is (A − A′)/2. Adding them returns A.

What does uniqueness of the inverse establish?

If a square matrix has an inverse, two matrices satisfying the inverse definition must be equal. The theorem does not establish existence for every square matrix.