Question

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. If A = `[(2, 0),(-1, 7)], I = [(1, 0),(0, 1)]` and A² = 9A + mI, then m = 14.
2. If A = `[(4, 0),(1, 2)], B = [(2),(3)]` and AX = B, then order of matrix X will be 2 × 2.

विकल्प

• Only (i)

• Only (ii)

• Both (i) and (ii)

• Neither (i) nor (ii)

Answer 1

Neither (i) nor (ii)

Explanation:

(i)

A = `[(2, 0),(-1, 7)]`

I = `[(1, 0),(0, 1)]`

A² = `[(2, 0),(-1, 7)] xx [(2, 0),(-1, 7)]`

= `[(2 xx 2 + 0 xx (-1), 2 xx 0 + 0 xx 7),(-1 xx 2 + 7 xx -1, -1 xx 0 + 7 xx 7)]`

= `[(4, 0),(-2 - 7, 49)]`

= `[(4, 0),(-9, 49)]`

9A = `9 [(2, 0),(-1, 7)]`

= `[(18, 0),(-9, 63)]`

A² = 9A + mI

A² − 9A = mI

`[(4, 0),(-9, 49)] - [(18, 0),(-9, 63)] = m[(1, 0),(0, 1)]`

`[(4 - 18, 0 - 0),(-9 - (-9), 49 - 63)] = [(m, 0),(0, m)]`

`[(-14, 0),(0, -14)] = [(m, 0),(0, m)]`

m = −14

Therefore, statement (i) is invalid.

(ii)

A = `[(4, 0),(1, 2)], B = [(2),(3)]`

AX = B

Order of A = 2 × 2

Order of B = 2 × 1

A_{m × n} × X_{n × p} = B_m×p

A is 2 × 2 means m = 2, n = 2

B is 2 × 1 means m = 2, p = 1

So X must be:

n × p = 2 × 1

X = 2 × 1

Therefore, statement (ii) is invalid.