Matrices | ICSE Class 10 Maths Notes

This note covers matrices and their entries, order, row and column matrices, equality, null and identity matrices, compatibility for operations, addition and subtraction of 2 × 2 matrices, multiplication by a non-zero rational number, matrix multiplication and simple matrix equations.

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

Definition: A matrix is an ordered rectangular arrangement of numbers or functions. Each number or function in the arrangement is called an element or entry. A function is a rule assigning an output to each allowed input; the calculations here use numerical entries.

A row runs horizontally across a matrix. A column runs vertically down it. Capital letters such as A name whole matrices. The arrangement matters: the position of an entry is part of the information represented by the matrix.

The order records the number of rows followed by the number of columns. If m is the number of rows and n is the number of columns, the order is m × n, read as “m by n”. Both counts are positive integers.

A positive integer is a whole number greater than zero. A matrix of order m × n contains mn entries, where mn means m multiplied by n. Thus order describes the arrangement, whereas the product of the two counts gives the total number of entries.

How does an entry's position identify it?

Write aᵢⱼ for the entry in row i and column j of matrix A. The small labels i and j are indices: i identifies the row and j the column. For a 2 × 2 matrix, its entries are a₁₁, a₁₂, a₂₁ and a₂₂.

We shall also use compact bracket notation: [a₁₁ a₁₂; a₂₁ a₂₂]. Spaces separate entries within a row, and the semicolon separates the first row from the second. When an entry contains a calculation, round brackets group its terms. Read each row from left to right, then move down to the next row.

Worked example 1. A matrix contains 8 entries. Find all its possible orders.

Answer: The row count multiplied by the column count must equal 8. The possible ordered pairs are (1, 8), (8, 1), (4, 2) and (2, 4). Therefore the orders are 1 × 8, 8 × 1, 4 × 2 and 2 × 4.

The word ordered is important here. Interchanging the row and column counts changes the order unless those counts are equal. Knowing only the total number of entries does not, in general, identify one particular arrangement.

How do row, column and square matrices differ?

A row matrix has exactly one row. Its order is 1 × n, where n counts its columns. A column matrix has exactly one column. Its order is m × 1, where m counts its rows.

A square matrix has equal numbers of rows and columns. Its order can be written n × n, or described as a square matrix of order n. A 2 × 2 matrix is therefore square and contains four entries arranged in two rows.

What does a change in arrangement mean?

The row matrix [0 1] and the column matrix [0; 1] contain the same two numbers. Their orders differ: the first is 1 × 2 and the second is 2 × 1. The arrangement must therefore be recorded along with the entries.

Do not identify a type by the values alone. A row matrix can contain zero, positive or negative entries. What makes it a row matrix is its single row. The same distinction between arrangement and values applies to column and square matrices.

In a square matrix, the main diagonal consists of entries whose row and column numbers match. For a 2 × 2 matrix, these are a₁₁ at the top left and a₂₂ at the bottom right. The other two entries are outside this diagonal and are called off-diagonal entries.

This diagonal will distinguish the identity matrix from the null matrix. First establish that the matrix is square, then inspect the diagonal entries and the entries outside it. Counting rows and columns remains the starting point for identifying its type.

When are two matrices equal?

Equal matrices have the same order and equal entries in corresponding positions. Corresponding positions mean the same row number and the same column number. Both conditions are needed; matching some entries or merely having the same number of entries is insufficient.

If A and B name two equal matrices, write A = B. If aᵢⱼ is an entry of A and bᵢⱼ is the entry in the same position in B, equality requires aᵢⱼ = bᵢⱼ at every position.

For example, [2 3; 0 1] equals [2 3; 0 1]. However, [3 2; 0 1] does not equal [2 3; 0 1]. Their orders agree, but the top-left and top-right entries do not match their corresponding entries.

How do equal matrices give equations?

When entries contain unknown numbers, compare them position by position. Each corresponding pair gives an ordinary equation. Solve these equations together, then substitute the answers into every position to check that the whole matrix equality holds.

Worked example 2. Find the unknown numbers a, b, c and d if [(2a + b) (a − 2b); (5c − d) (4c + 3d)] = [4 −3; 11 24].

Answer: The top row gives 2a + b = 4 and a − 2b = −3. Doubling the first equation and adding the second gives 5a = 5, so a = 1 and b = 2.

The bottom row gives 5c − d = 11 and 4c + 3d = 24. Multiplying the first of these by 3 and adding the second gives 19c = 57. Hence c = 3 and d = 4.

For this example, the upper entries become 2 × 1 + 2 = 4 and 1 − 2 × 2 = −3. The lower entries become 5 × 3 − 4 = 11 and 4 × 3 + 3 × 4 = 24.

The check confirms all four positions. The four unknowns do not belong to one long numerical list: they occur in entries whose positions determine which equations to form. Retaining the row layout helps prevent accidental comparison of neighbouring entries.

What are null and identity matrices?

A null matrix, also called a zero matrix, has zero in every position. We use the capital letter O for a null matrix, with its order stated or clear from the calculation. A null matrix need not be square.

The 2 × 2 null matrix is O = [0 0; 0 0]. By contrast, a matrix with just one zero entry is not necessarily null. The definition concerns every entry, so check the complete arrangement rather than a single row or diagonal.

An identity matrix is square, with 1 in every main-diagonal position and 0 in every other position. Write I for the identity matrix when its order is clear. Here the 2 × 2 identity matrix is I = [1 0; 0 1].

Property: The null matrix is the additive identity

An additive identity leaves a matrix unchanged when added to it. For any matrix A and a null matrix O of the same order, A + O = O + A = A. Each entry has zero added to it.

Property: The identity matrix is the multiplicative identity

A multiplicative identity leaves a matrix unchanged when used in multiplication. For a square matrix A and identity matrix I of the same order, AI = IA = A. Here AI means the matrix product of A followed by I.

These names refer to different operations. Adding the identity matrix is not the same as multiplying by it. When using either property, state the order so that the matrix operation has a definite meaning and the entries can be matched correctly.

How do you check whether matrix operations are compatible?

Compatibility means that the matrices satisfy the size conditions needed for an operation to be defined. Check it before calculating any entries. Addition and multiplication use different conditions, so a pair can be suitable for one operation without being suitable for another.

Addition and subtraction require matrices of the same order. This provides a corresponding entry for every position. Equal totals of entries are not enough if the numbers of rows and columns differ.

For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second. If A has order m × n and B has order n × p, where p counts B's columns, then AB has order m × p.

Which counts determine the answer's order?

The matching inner counts allow the product to be calculated. The two outer counts give the resulting order: rows from the first matrix and columns from the second. This tells you how many answer positions to prepare before doing the arithmetic.

Worked example 3. Matrix A has order 2 × 2 and matrix B has order 2 × 3. Decide whether AB and BA are defined, and give the order of any defined product.

Answer: AB is defined because A has 2 columns and B has 2 rows. It has order 2 × 3. BA is not defined because B has 3 columns, whereas A has 2 rows.

Here A and B also cannot be added: their orders differ. This decision uses their sizes and does not require the numerical entries. A question about compatibility can therefore supply orders alone, while a question asking for a numerical product must supply the entries.

Note: If AB is defined, BA need not be defined. Check the reversed order separately. When both matrices are square and of the same order, both products are defined, but they need not be equal.

How do you add two 2 × 2 matrices?

To add two matrices of order 2 × 2, add their corresponding entries. The answer retains that order. The top-left sum stays at the top left, and the same position rule applies to each of the other three sums.

Let A = [a₁₁ a₁₂; a₂₁ a₂₂] and B = [b₁₁ b₁₂; b₂₁ b₂₂], where the small labels identify row and column positions in their respective matrices. Then A + B = [(a₁₁ + b₁₁) (a₁₂ + b₁₂); (a₂₁ + b₂₁) (a₂₂ + b₂₂)].

1. Check that both matrices have two rows and two columns.
2. Add the two top-left entries and place the result at the top left.
3. Add the top-right entries, then the bottom-left entries, retaining their positions.
4. Add the bottom-right entries and write the four sums inside one pair of matrix brackets.

Worked example 4. Given X = [4 4; 0 4] and Y = [1 −2; 0 5], find X + Y.

Answer: Both matrices have order 2 × 2. Add entry by entry: 4 + 1 = 5, 4 + (−2) = 2, 0 + 0 = 0 and 4 + 5 = 9. Therefore X + Y = [5 2; 0 9].

Property: Addition is commutative and associative

Commutative means that reversing the order does not change the sum. For matrices A and B of the same order, A + B = B + A. This follows because the addition of each pair of numerical entries is commutative.

Associative means that changing the grouping does not change the sum. If C is a third matrix of that same order, then (A + B) + C = A + (B + C). Brackets specify which addition is performed first.

Both properties retain the requirement about order. They describe sums that are defined; they do not make incompatible matrices addable. For a calculation involving several matrices, establish a common order before changing either the grouping or the sequence of additions.

How do you subtract matrices without losing negative signs?

For two matrices of the same order, subtract each entry of the second matrix from the corresponding entry of the first. Thus A − B means the first matrix A minus the second matrix B, with every answer kept in its original position.

The negative of a matrix, written −B, is obtained by changing the sign of every entry of B. Matrix subtraction can then be written as A − B = A + (−B). This is useful when the second matrix already contains negative entries.

How does subtracting a negative entry work?

Retain brackets around a negative entry during substitution. An expression such as 4 − (−2) becomes 4 + 2. The subtraction sign belongs to the operation, while the negative sign inside the brackets belongs to the entry being subtracted.

Worked example 5. Given X = [4 4; 0 4] and Y = [1 −2; 0 5], find X − Y.

Answer: Both matrices have order 2 × 2. The entries of the difference are 4 − 1 = 3, 4 − (−2) = 6, 0 − 0 = 0 and 4 − 5 = −1. Hence X − Y = [3 6; 0 −1].

Compare this answer with the sum of the same matrices. The operations use corresponding positions in both cases, but subtraction changes the arithmetic at every position. Do not reuse an addition result without reconsidering the operation requested.

The additive inverse of A is −A: adding these two matrices gives the null matrix of the same order. Symbolically, A + (−A) = O. Each entry cancels with its own negative, which also explains why subtracting a matrix from itself gives a null matrix.

When subtracting a bracketed matrix expression, apply the minus sign to the whole expression. In particular, removing brackets from X − Y under a preceding subtraction changes both signs. Writing the intermediate entry calculations makes these sign changes visible.

How do you multiply a matrix by a non-zero rational number?

A scalar is a single number used to multiply a matrix. A rational number can be written as a fraction with integer numerator and non-zero integer denominator. Integers include negative whole numbers, zero and positive whole numbers. Non-zero means the number is not zero.

If k denotes a non-zero rational scalar and A is a matrix, kA is obtained by multiplying every entry of A by k. The order stays unchanged. The numerator is the number above a fraction bar; the denominator is the number below it.

For a 2 × 2 matrix A = [a₁₁ a₁₂; a₂₁ a₂₂], scalar multiplication gives kA = [ka₁₁ ka₁₂; ka₂₁ ka₂₂]. Each juxtaposition, such as ka₁₁, means multiplication of the scalar and that entry.

Worked example 6. Let M = [8 8; 0 8]. Find ½M, where ½ means one half.

Answer: Multiply all four entries by ½. Since ½ × 8 = 4 and ½ × 0 = 0, the result is ½M = [4 4; 0 4]. The matrix remains of order 2 × 2.

How does scalar multiplication interact with addition?

For same-order matrices A and B, k(A + B) = kA + kB. This is a distributive property: multiplying a sum gives the same result as multiplying each matrix separately and then adding the results.

If l denotes another scalar, then (k + l)A = kA + lA. These rules work entry by entry. They help simplify matrix equations because a numerical factor applies to the entire matrix, including entries containing brackets or negative signs.

Multiplication by −1 produces the negative matrix: −A = (−1)A. Multiplication by a fraction also applies to every entry. In an equation with 2X on one side, multiplying that whole matrix by ½ isolates the unknown matrix X.

Note: A scalar outside a matrix bracket acts on every entry. If an entry itself contains a difference, keep that difference bracketed until the scalar has multiplied both its terms.

How does row-by-column matrix multiplication work?

A matrix product combines rows of the first matrix with columns of the second. For each answer entry, multiply corresponding numbers in the chosen row and column, then add the products. This differs from simply multiplying entries occupying identical positions.

For two 2 × 2 matrices, every answer entry contains a sum of two numerical products. There are four answer positions. The first matrix supplies the row for each calculation, and the second supplies the column.

How is each product entry formed?

Let A = [a b; c d] and B = [e f; g h], where a, b, c, d, e, f, g and h name their numerical entries in the displayed positions. Then AB = [(ae + bg) (af + bh); (ce + dg) (cf + dh)].

1. Use the first row of A and the first column of B for the top-left entry: ae + bg.
2. Use the first row of A and the second column of B for the top-right entry: af + bh.
3. Use the second row of A and the first column of B for the bottom-left entry: ce + dg.
4. Use the second row of A and the second column of B for the bottom-right entry: cf + dh.

Worked example 7. Prices are in rupees, written using the symbol ₹. Meera needs 2 pens and 5 story books; Nadeem needs 8 pens and 10 story books. A pen and a story book cost ₹5 and ₹50 at one shop, and ₹4 and ₹40 at another. Find both people's costs at both shops.

Answer: Let R = [2 5; 8 10] represent requirements, with Meera's row first. Let P = [5 4; 50 40] represent prices, with pens in the first row and the first shop in the first column.

Multiply to obtain RP = [(2 × 5 + 5 × 50) (2 × 4 + 5 × 40); (8 × 5 + 10 × 50) (8 × 4 + 10 × 40)] = [260 208; 540 432].

Meera needs ₹260 at the first shop and ₹208 at the second. Nadeem needs ₹540 at the first shop and ₹432 at the second. Rows identify people and columns identify shops in the product.

The labels explain why the terms are added: each person's total includes the cost of pens and the cost of story books. Matching the item order in the requirements and price matrices ensures that each quantity multiplies its correct price.

A 2 × 2 matrix can also multiply a compatible column matrix. Using only the first shop's prices gives R[5; 50] = [260; 540]. The product has one column because the price matrix has one column; its two rows still correspond to the two people.

Which multiplication rules need special care?

Matrix multiplication is not commutative in general. This means that reversing two matrices can change their product. Even when both products exist and have the same order, their entries need not agree. The order of the factors must be preserved.

Can reversing the factors change the answer?

Worked example 8. Let A = [1 0; 0 −1] and B = [0 1; 1 0]. Find AB and BA.

Answer: Row-by-column multiplication gives AB = [0 1; −1 0], whereas BA = [0 −1; 1 0]. The top-right entries differ, so AB ≠ BA, where ≠ means “is not equal to”. Both products nevertheless have order 2 × 2.

This does not mean that reversed products differ for every pair. For A = [1 0; 0 2] and B = [3 0; 0 4], both AB and BA equal [3 0; 0 8]. Thus equality must be established for the particular matrices.

Can two non-null matrices have a null product?

Worked example 9. Find AB for A = [0 −1; 0 2] and B = [3 5; 0 0].

Answer: The entries are 0 × 3 + (−1) × 0 = 0, 0 × 5 + (−1) × 0 = 0, 0 × 3 + 2 × 0 = 0 and 0 × 5 + 2 × 0 = 0. Therefore AB = [0 0; 0 0] = O, although neither factor is null.

A non-null matrix has at least one non-zero entry. The example shows why a null product does not establish that a factor is null. This familiar implication for ordinary numerical multiplication does not hold for matrices in general.

Property: Multiplication is associative and distributive

For matrices A, B and C, (AB)C = A(BC) whenever both sides are defined. This associative rule changes the grouping, while keeping the factors in the same sequence.

The distributive rules are A(B + C) = AB + AC and (A + B)C = AC + BC, whenever both sides are defined. Keep the multiplying matrix on its original side of each term. These rules do not authorise interchanging factors.

How can matrix operations solve simple matrix equations?

A matrix equation states that two matrix expressions are equal. An unknown may be an entry or a whole matrix. Use compatible addition, subtraction and scalar multiplication to simplify the equation, then compare corresponding positions where needed.

How do you find unknown entries?

Worked example 10. Find the unknown numbers x and y if 2[x 5; 7 (y − 3)] + [3 −4; 1 2] = [7 6; 15 14].

Answer: Multiplying every entry of the first matrix by 2 and adding gives [(2x + 3) 6; 15 (2y − 4)] = [7 6; 15 14]. Equality requires 2x + 3 = 7 and 2y − 4 = 14. Therefore x = 2 and y = 9.

The two entries without unknowns also agree: 10 − 4 = 6 and 14 + 1 = 15. These are part of checking the full equation. The expression y − 3 must be doubled as a whole before the entry 2 is added.

How do you recover matrices from their sum and difference?

Worked example 11. Find the 2 × 2 matrices X and Y if X + Y = [5 2; 0 9] and X − Y = [3 6; 0 −1].

Answer: Adding the equations cancels Y and gives 2X = [8 8; 0 8]. Multiplying each entry by ½ gives X = [4 4; 0 4].

Subtracting the second equation from the first cancels X and gives 2Y = [2 −4; 0 10]. Hence Y = [1 −2; 0 5]. Their sum and difference reproduce the two given matrices.

This method uses the same-order condition throughout. The left sides combine by addition rules; the right sides combine entry by entry. The factor ½ then applies to every entry, so the recovered matrices retain order 2 × 2.

A useful final check is to substitute the complete answers into the original equations. Checking only one entry can miss a misplaced value or sign elsewhere. For two unknown matrices, check both their given sum and their given difference.

Glossary

• Matrix — An ordered rectangular arrangement of numbers or functions, with each entry occupying a specific position.
• Entry — An individual number or function occupying a particular row and column position in a matrix.
• Order — The size of a matrix, written as its row count followed by its column count.
• Row matrix — A matrix with exactly one row and one or more columns.
• Column matrix — A matrix with exactly one column and one or more rows.
• Square matrix — A matrix whose number of rows equals its number of columns.
• Main diagonal — Entries in a square matrix whose row and column numbers are equal.
• Null matrix — A matrix in which every entry is zero, also called a zero matrix.
• Identity matrix — A square matrix with ones on its main diagonal and zeros elsewhere.
• Equal matrices — Matrices of the same order with equal entries at every corresponding position.
• Scalar — A single number that multiplies every entry when used to multiply a matrix.
• Compatibility — Satisfaction of the size conditions required for a particular matrix operation to be defined.
• Additive inverse — The negative of a matrix, which gives a null matrix when added to the original.
• Matrix product — The result of combining rows of the first matrix with columns of the second by multiplication and addition.

Common errors and misconceptions

• Misconception: Order lists columns before rows. Correct: Order lists rows first and columns second; entry indices follow this same sequence.
• Misconception: Equal numbers of entries make matrices addable. Correct: Addition requires the same order, providing a corresponding entry at every position.
• Misconception: Any matrix containing zero is null. Correct: Every entry must be zero for the matrix to be null.
• Misconception: A scalar outside brackets affects only the first entry. Correct: It multiplies every entry, including every term inside an entry's own brackets.
• Misconception: Matrix multiplication multiplies corresponding entries. Correct: Each product entry is a sum of products formed from a row and a column.
• Misconception: Reversing matrix factors must preserve their product. Correct: The reversed product may be undefined or different, though some pairs do give equal products.
• Misconception: A null product means at least one factor is null. Correct: Two non-null matrices can have a null product.
• Misconception: Subtracting a negative entry keeps it negative. Correct: Subtracting a negative is addition; retain brackets while applying the subtraction sign.

Exam-style questions with model answers

Q1. A matrix has 8 entries. State all possible orders and explain the condition you use. [2 marks]

1. If m counts rows and n counts columns, both are positive integers and their product mn must equal 8.
2. The possible orders are 1 × 8, 8 × 1, 4 × 2 and 2 × 4.

Q2. Matrix A has order 2 × 2 and matrix B has order 2 × 3. Decide whether A + B, AB and BA are defined, giving reasons and the order of any defined product. [3 marks]

1. A + B is not defined because the matrices have different orders. Addition requires both the same row count and the same column count.
2. AB is defined because A's two columns match B's two rows. Its order is 2 × 3, using A's rows and B's columns.
3. BA is not defined because B has three columns but A has only two rows. The inner counts do not match.

Q3. Given X = [4 4; 0 4] and Y = [1 −2; 0 5], calculate X + Y and X − Y, showing the entry calculations. A semicolon separates rows. [4 marks]

1. Both matrices have order 2 × 2, so their sum and difference are defined. Corresponding entries must remain in the same positions.
2. For the sum, the top entries are 4 + 1 = 5 and 4 + (−2) = 2. The bottom entries are 0 + 0 = 0 and 4 + 5 = 9. Thus X + Y = [5 2; 0 9].
3. For the difference, the top entries are 4 − 1 = 3 and 4 − (−2) = 6.
4. The bottom entries are 0 − 0 = 0 and 4 − 5 = −1. Therefore X − Y = [3 6; 0 −1].

Q4. Prices are in rupees, denoted by ₹. Meera needs 2 pens and 5 story books; Nadeem needs 8 pens and 10 story books. Pens and story books cost ₹5 and ₹50 each at the first shop, and ₹4 and ₹40 each at the second. Use matrix multiplication to find each person's cost at each shop. [5 marks]

1. Write the requirements matrix R = [2 5; 8 10]. Rows represent Meera and Nadeem, respectively; columns represent pens and story books, respectively.
2. Write the price matrix P = [5 4; 50 40]. Its rows represent pens and story books, while its columns represent the first and second shops.
3. Both matrices have order 2 × 2, so RP is defined and has order 2 × 2. Use rows of R with columns of P.
4. Meera's costs are 2 × 5 + 5 × 50 = ₹260 and 2 × 4 + 5 × 40 = ₹208.
5. Nadeem's costs are 8 × 5 + 10 × 50 = ₹540 and 8 × 4 + 10 × 40 = ₹432. Hence RP = [260 208; 540 432], with rows for people and columns for shops.

Q5. Let A = [1 0; 0 −1] and B = [0 1; 1 0], with semicolons separating rows. Calculate AB and BA and explain what the comparison shows. [4 marks]

1. Both matrices have order 2 × 2, so both products exist and have order 2 × 2. Use row-by-column multiplication in each case.
2. For AB, the first row is [0 1] and the second row is [−1 0]. Hence AB = [0 1; −1 0].
3. For BA, the first row is [0 −1] and the second row is [1 0]. Hence BA = [0 −1; 1 0].
4. The corresponding off-diagonal entries differ, so AB is not equal to BA. Matrix multiplication is not commutative in general; this does not assert that reversed products differ for every pair.

Q6. Find the 2 × 2 matrices X and Y if X + Y = [5 2; 0 9] and X − Y = [3 6; 0 −1]. Show your operations and check both answers. A semicolon separates rows. [5 marks]

1. Add the given equations. The Y terms cancel, leaving 2X = [(5 + 3) (2 + 6); (0 + 0) (9 + (−1))] = [8 8; 0 8].
2. Multiply every entry by ½, meaning one half. This gives X = [4 4; 0 4], still of order 2 × 2.
3. Subtract the second equation from the first. The X terms cancel and the negative Y changes sign, giving 2Y = [2 −4; 0 10].
4. Multiply every entry by ½ to obtain Y = [1 −2; 0 5]. The top-right entry is negative because half of −4 is −2.
5. Check the results: X + Y = [5 2; 0 9] and X − Y = [3 6; 0 −1]. Both original equations are satisfied entry by entry.

Key takeaways

• A matrix's order gives its rows before its columns; their product counts all the entries.
• Two matrices are equal when their orders agree and every pair of corresponding entries is equal.
• Null matrices contain zeros throughout; identity matrices have ones on the main diagonal and zeros elsewhere.
• Addition and subtraction require equal orders and operate on corresponding entries without changing their positions.
• A non-zero rational scalar multiplies every entry of a matrix while preserving its order.
• Matrix multiplication requires matching inner counts and forms each answer entry from a row and a column.
• Reversed matrix products need not be defined or equal, and non-null factors can produce a null matrix.
• To recover two matrices from their sum and difference, add or subtract the equations, then halve every entry.

Test yourself

What do the two indices in aᵢⱼ identify?

The index i identifies the row, and j identifies the column containing that entry of matrix A.

How do the orders of [0 1] and [0; 1] differ, with the semicolon separating rows?

The row matrix has order 1 × 2; the column matrix has order 2 × 1.

Write the null and identity matrices of order 2 × 2.

The null matrix is [0 0; 0 0]. The identity matrix is [1 0; 0 1], with semicolons separating rows.

What condition allows a 2 × 2 matrix to multiply a second matrix?

The second matrix must have two rows, matching the two columns of the first matrix.

What does multiplying [8 8; 0 8] by ½ produce? A semicolon separates rows.

Multiplying every entry by one half gives [4 4; 0 4], with the order unchanged.

If AB is defined, must BA be defined too?

No. The reversed product requires its own matching column and row counts, which may not agree.

If two matrices have a null product, must one of them be null?

No. Two non-null matrices can have a null product, so the product alone does not establish a null factor.

Why does adding equations for X + Y and X − Y help find X?

The Y terms cancel, leaving 2X. Multiplying every entry of that resulting matrix by one half gives X.