If \(A = \begin{bmatrix}1&-1&2\\0&3&1\\2&1&0\end{bmatrix}\), then the sum of the diagonal elements of \(A^2\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 20
Only the diagonal elements of \(A^2\) are needed. The element in row i and column i is found from row i of A and column i of A.
Row 1 with column 1: \(1(1) + (-1)(0) + 2(2) = 5\)
Row 2 with column 2: \(0(-1) + 3(3) + 1(1) = 10\)
Row 3 with column 3: \(2(2) + 1(1) + 0(0) = 5\)
Hence the sum of the diagonal elements of \(A^2\) is
\(5 + 10 + 5 = 20\)
Why the other options are incorrect
Option (1) 10 — Incorrect. This is \(1^2 + 3^2 + 0^2\), the squares of the diagonal elements of A only; the diagonal of \(A^2\) also uses the other elements.
Option (2) 16 — Incorrect. This is \((1 + 3 + 0)^2\), the square of the diagonal sum of A.
Option (3) 14 — Incorrect. The three diagonal elements of \(A^2\) are 5, 10 and 5, which add to 20.