If A = [(3,1)(7,5)] , find x and y such that A^2 + xI = yA.

Question

If A = \(\begin{bmatrix} 3 &1 \\[0.3em] 7 & 5 \\[0.3em] \end{bmatrix}\), find x and y such that A² + xI = yA.

A = [(3,1)(7,5)]

Answer 1

Given : A = \(\begin{bmatrix} 3& 1 \\[0.3em] 7 &5 \\[0.3em] \end{bmatrix}\), A² + xI = yA.

A is a matrix of order 2 x 2

To find : x and y

Formula used :

Where c_ij = a_i1b_1j + a_i2b₂j + 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

A² is a matrix of order 2 x 2

It is given that A² + xI = yA,

Equating similar terms in the given matrices,

16 + x = 3y and 8 = y,

hence y = 8

Substituting y = 8 in equation 16 + x = 3y

16 + x = 3 × 8 = 24 

16 + x = 24 

x = 24 – 16 = 8 

x = 8 

x = 8, y = 8