MathematicsClass 10ICSE

Why A Times B Is Usually Not the Same as B Times A for Matrices

Learn when two matrices can be multiplied and what order the product has, multiply 2x2 and column matrices, verify that matrix multiplication is not commutative but is distributive, and solve AX = B and A² = kA + mI.

How is multiplying matrices different from adding them?

Adding matrices works element by element. Multiplication does not. Instead, each row of the first matrix is combined with each column of the second: multiply matching elements and add.

[a c b d] [p r q s] = [a p + b r c p + d r a q + b s c q + d s]

This row-into-column rule makes the order of multiplication matter. This part covers when multiplication is possible, how to multiply, its properties and matrix equations.

When can two matrices be multiplied, and what is the order of the product?

**A matrix of order

m \times n

can be multiplied by one of order

p \times q

only when

n = p

, and the product then has order

m \times q

.

Worked examples.**

-

2 \times 2

2 \times 1

— possible; product is

2 \times 1

2 \times 1

2 \times 2

— not possible, since

1 \neq = 2

1 \times 2

2 \times 1

— possible; product is

1 \times 1

2 \times 1

1 \times 2

— possible; product is

2 \times 2

An everyday example. **Buying

3

pens at ₹10 and

5

notebooks at ₹40**, the bill is a row of quantities times a column of prices:

[35] [1040] = [3 \times 10 + 5 \times 40] = [230]

The total is ₹230.

The substance. **

A B

can exist while

B A

does not**, because the inner numbers of the orders must match.

How do you multiply a 2x2 matrix by another 2x2 matrix and by a column matrix?

Multiply each row of the first matrix by each column of the second, adding the products of matching elements, and place each result in the row and column it came from.

Worked example 1. Let

A = [2314], B = [10 - 2 5]

A B = [2 (1) + 1 (0) 3 (1) + 4 (0) 2 (- 2) + 1 (5) 3 (- 2) + 4 (5)] = [23114]

Worked example 2 — column matrix.

A [3 - 1] = [2 (3) + 1 (- 1) 3 (3) + 4 (- 1)] = [55]

An everyday example. Two kirana shops charge per kg ₹50 and ₹120 for rice and dal, and ₹48 and ₹125. For

5

kg rice and

2

kg dal:

[5048120125] [52] = [490490]

Both shops come to ₹490.

The substance. The identity matrix leaves a matrix unchanged:

A I = I A = A

, just as multiplying a number by

1

does.

How do you verify that matrix multiplication is not commutative but is distributive over addition?

**Compute

A B

and

B A

for the same matrices to see they differ, and compute

A (B + C)

and

A B + A C

to see they agree.

Not commutative.** With

A

and

B

above:

B A = [1 (2) + (- 2) (3) 0 (2) + 5 (3) 1 (1) + (- 2) (4) 0 (1) + 5 (4)] = [- 4 15 - 7 20] \neq = A B

Distributive. Let

C = [12 0 - 1]

, so

B + C = [22 - 2 4]

A (B + C) = [2 (2) + 1 (2) 3 (2) + 4 (2) 2 (- 2) + 1 (4) 3 (- 2) + 4 (4)] = [614010]

A C = [411 - 1 - 4] A B + A C = [2 + 4 3 + 11 1 - 1 14 - 4] = [614010]

An everyday example. Adding tadka before or after boiling the dal gives different results — some operations depend on order, just like matrix products.

The substance. **Because

A B \neq = B A

(A + B)^{2} = A^{2} + A B + B A + B^{2}

**, not

A^{2} + 2 A B + B^{2}

How do you solve matrix equations such as AX = B or A² = kA + mI?

Write the unknown matrix with letters, carry out the multiplication, and equate corresponding elements to get ordinary equations.

**Worked example 1 —

A X = B

.** Solve

[2314] [x y] = [55]

2 x + y = 53 x + 4 y = 5

From the first,

y = 5 - 2 x

. Then

3 x + 20 - 8 x = 5

, so

x = 3

and

y = - 1

— matching the column product found earlier.

**Worked example 2 —

A^{2} = k A + m I

.** For

A = [2314]

A^{2} = [4 + 3 6 + 12 2 + 4 3 + 16] = [718619] k A + m I = [2 k + m 3 k k 4 k + m]

k = 6, 3 k = 18, 2 k + m = 7 \Rightarrow m = - 5, 4 k + m = 24 - 5 = 19

**So

A^{2} = 6 A - 5 I

.

An everyday example. A shopkeeper who has lost a price list** can recover two unknown prices from two customers' bills, written as

A X = B

.

The substance. Check all four element equations — the unused ones confirm the values are consistent.

Exam tip

What earns full marks on matrix multiplication?

Check compatibility first, show every row-column product, and equate all corresponding elements in matrix equations.

- Write orders and confirm the inner numbers match
- State the order of the product before calculating
- Show each element as a sum of products
- **Never assume

A B = B A

- In

A^{2} = k A + m I

**, use all four elements to find and check

k

and

m

The trap. Multiplying element by element as in addition. **

A B

uses rows of

A

with columns of

B

**, never matching positions.

Did you know

Can two non-zero matrices multiply to give the zero matrix?

With numbers, if

ab = 0

, then

a

b

must be

0

. Matrices break this rule.

[1111] [1 - 1 - 1 1] = [1 - 1 1 - 1 - 1 + 1 - 1 + 1] = [0000]

Neither matrix is zero, yet the product is. That is why you cannot cancel matrices in an equation the way you cancel numbers.

Exam relevance

How is matrix multiplication tested in JEE Main?

This is foundation work for Class 12 Matrices and Determinants, both JEE Main chapters.

What gets built on. Class 12 uses multiplication to define inverses, prove properties such as the transpose of a product, and solve systems of equations as

A X = B

. The result

A^{2} = 6 A - 5 I

above is no accident: **every

2 \times 2

matrix satisfies an equation of this form**, where

6

is the sum of the diagonal elements and

5

is the determinant

2 \times 4 - 1 \times 3

.

Question types. Multiple-choice and numerical-value questions on powers of matrices, products and unknown elements.

The trap that costs marks. **Assuming

A B = B A

** when expanding expressions such as

(A + B)^{2}

Key takeaways

What must you be able to do from this part?

- **

m \times n

times

p \times q

** is possible only if

n = p

; the product is

m \times q

- Row into column: multiply matching elements and add
- **

A B = [23114]

** but

B A = [- 4 15 - 7 20]

in the example
- Distributive:

A (B + C) = A B + A C

- **

A X = B

**: equate elements;

x = 3

y = - 1

- **

A^{2} = k A + m I

**:

k = 6

m = - 5

- Non-zero matrices can multiply to give the zero matrix

Pick any

2 \times 2

matrix, square it, and see whether you can find

k

and

m

so that

A^{2} = k A + m I

How is multiplying matrices different from adding them?

When can two matrices be multiplied, and what is the order of the product?

How do you multiply a 2x2 matrix by another 2x2 matrix and by a column matrix?

How do you verify that matrix multiplication is not commutative but is distributive over addition?

How do you solve matrix equations such as AX = B or A² = kA + mI?

What earns full marks on matrix multiplication?

Can two non-zero matrices multiply to give the zero matrix?

How is matrix multiplication tested in JEE Main?

What must you be able to do from this part?

Related study notes

Ready to put this into practice?