Question

If A= `[(-4, 2),(5, -1)], B = [(17, -1),(47, -13)]` and CA = B, then matrix C is ______.

पर्याय

• `[(2, -5),(-3, 7)]`

• `[(-2, 5),(-3, 7)]`

• `[(2, 5),(-3, 7)]`

• `[(2, 5),(3, 7)]`

Answer 1

If A= `[(-4, 2),(5, -1)], B = [(17, -1),(47, -13)]` and CA = B, then matrix C is `underlinebb([(2, 5),(-3, 7)])`.

Explanation:

Let C = `[(a, b),(c, d)]`

CA = `[(a, b),(c, d)] xx [(-4, 2),(5, -1)]`

= `[(a(-4) + b(5), a(2) + b(-1)),(c(-4) + d(5), c(2) + d(-1))]`

= `[(-4a + 5b, 2a - b),(-4c + 5d, 2c - d)]`

Set equal to B:

`[(-4a + 5b, 2a - b),(-4c + 5d, 2c - d)] = [(17, -1),(47, -13)]`

Compare corresponding elements

−4a + 5b = 17   ...(1)

2a − b = −1   ...(2)

Solve (2) for b

b = 2a + 1

Substitute into (1):

−4a + 5b = 17

−4a + 5(2a + 1) = 17

−4a + 10a + 5 = 17

6a = 17 −5

6a = 12

a = `12/6`

a = 2

Now,

b = 2a + 1

= 2(2) + 1

= 4 + 1

= 5

from lower row:

−4c + 5d = 47   ...(3)

2c − d =−13   ...(4)

Substitute into (3):

−4c + 5d = 47

−4c + 5(2c + 13) = 47

−4c + 10c + 65 = 47

6c = 47 − 65

6c = −18

c = `−18/6`

c = −3

Then,

d = 2c + 13

= 2(−3) + 13

= −6 + 13

= 7

C = `[(a, b),(c, d)] = [(2, 5),(-3, 7)]`