Find the values of a and b for which [(a,b)(-a,2b)][(2,-1)] = [(5,4)].

Question

Find the values of a and b for which

\( \begin{bmatrix} a& b \\[0.3em] -a&2b \\[0.3em] \end{bmatrix}\)\( \begin{bmatrix} 2 \\[0.3em] -1 \\[0.3em] \end{bmatrix}\) = \( \begin{bmatrix} 5 \\[0.3em] 4 \\[0.3em] \end{bmatrix}.\) 

[(a,b)(-a,2b)][(2,-1)] = [(5,4)].

Answer 1

Given : \( \begin{bmatrix} a& b \\[0.3em] -a&2b \\[0.3em] \end{bmatrix}\)\( \begin{bmatrix} 2 \\[0.3em] -1 \\[0.3em] \end{bmatrix}\) = \( \begin{bmatrix} 5 \\[0.3em] 4 \\[0.3em] \end{bmatrix}.\) 

To find : a and b

Formula used :

Where c_ij = a_i1b_1j + a_i2b_2j + a_i3b_3j + ……………… + a_inb_nj

If A is a matrix of order a x b and B is a matrix of order c x d ,then matrix AB exists and is of order a x d ,

if and only if b = c

Equating similar terms,

2a – b = 5

-2a – 2b = 4

Adding the above two equations,we get

-3b = 9

b = \(\frac{9}{-3}=-3\)

b = -3

substituting b = -3 in 2a – b = 5,we get

2a + 3 = 5

2a = 5 – 3 = 2

a = 1

a = 1 and b = -3