If A = [(1,0,-1),(2,1,3),(0,1,1)], then verify that A^2 + A = A(A + I), where I is 3 × 3 unit matrix.

Question

If A = \(\begin{bmatrix} 1 & 0 & -1 \\[0.3em] 2 & 1 & 3 \\[0.3em] 0 & 1 & 1 \end{bmatrix}\) , then verify that A² + A = A(A + I), where I is 3 × 3 unit matrix.

Answer 1

We are given with matrix A, such that

Multiply 1^st row of matrix A by matching members of 1^st column of matrix A, then sum them up.

(1, 0, -1)(1, 2, 0) = (1 × 1) + (0 × 2) + (-1 × 0)

⇒ (1, 0, -1)(1, 2, 0) = 1 + 0 + 0

⇒ (1, 0, -1)(1, 2, 0) = 1

Similarly, repeat steps to fill for the other elements.

Now, add A² and A,

Take R.H.S: A(A + I)

First, let us solve for (A + I).

Since, L.H.S = R.H.S.

Thus, (A² + A) = A(A + I).