If A = [(2,3)(1,2)]and I = [(1,0)(0,1)], then find λ, μ so that A^2 = λA + μI ​

Question

If A =\( \begin{bmatrix} 2 &3 \\[0.3em] 1 & 2 \end{bmatrix}\)and I =\( \begin{bmatrix} 1 &0 \\[0.3em] 0 & 1 \end{bmatrix},\) then find λ, μ so that A² = λA + μI

Answer 1

Given,

 A =\( \begin{bmatrix} 2 &3 \\[0.3em] 1 & 2 \end{bmatrix}\)and I =\( \begin{bmatrix} 1 &0 \\[0.3em] 0 & 1 \end{bmatrix}\) and  

A² = λA + μI

So,

[as r_ij = a_ij + b_ij + c_ij],

And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal 

Hence, 

λ + 0 = 4 ⇒ λ = 4 

And also, 

2λ + μ = 7 

Substituting the obtained value of λ in the above equation, we get 

2(4) + μ = 7 

⇒ 8 + μ = 7 

⇒ μ = – 1 

Therefore,

The value of λ and μ are 4 and – 1 respectively.