Question

In the given case below, find:

1. the order of matrix M.
2. the matrix M.

1. `M xx [(1, 1),(0, 2)] = [(1, 2)]`
2. `[(1, 4),(2, 1)] xx M = [(13), (5)]`

Answer 1

We know, the product of two matrices is defined only when the number of columns of first matrix is equal to the number of rows of the second matrix.

i. Let the order of matrix M be a × b.

`M_(a xx b) xx [(1, 1),(0, 2)]_(2 xx 2) = [(1, 2)]_(1 xx 2)`

Clearly, the order of matrix M is 1 × 2.

Let `M = [(a, b)]`

`M xx [(1, 1),(0, 2)] = [(1, 2)]`

`[(a, b)] xx [(1, 1),(0, 2)] = [(1, 2)]`

`[(a + 0, a + 2b)] = [(1, 2)]`

Comparing the corresponding elements, we get,

a = 1 and a + 2b = 2

`=>` 2b = 2 – 1 = 1

`=> b = 1/2`

∴ `M = [(a, b)] = [(1, 1/2)]`

ii. Let the order of matrix M be a × b.

`[(1, 4),(2, 1)]_(2 xx 2) xx M_(a xx b) = [(13),(5)]_(2 xx 1)`

Clearly, the order of matrix M is 2 × 1.

Let `M = [(a), (b)]`

`[(1, 4),(2, 1)] xx M = [(13),(5)]`

`[(1, 4),(2, 1)] xx [(a),(b)] = [(13),(5)]`

`[(a + 4b),(2a + b)] = [(13),(5)]`

Comparing the corresponding elements, we get,

a + 4b = 13  ...(1)

2a + b = 5   ...(2)

Multiplying (2) by 4, we get,

8a + 4b = 20  ...(3)

Subtracting (1) from (3), we get,

7a = 7

`=>` a = 1

From (2), we get,

b = 5 – 2a

= 5 – 2

= 3

∴ `M = [(a),(b)] = [(1),(3)]`