Matrices & Determinants
Matrix multiplication with parameters
nta_pyq_2025_apr
Grade 12
Question:
Let the matrix $A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$ satisfy $A^n = A^{n-2} + A^2 - I$ for $n \geq 3$. Then the sum of all the elements of $A^{50}$ is:
Step-by-Step Solution
Key Concept: Apply the matrix property for matrix multiplication with parameters and reduce it to determinant or parameter equations.
$A^{50} = A^{48} + A^2 - I$
$= A^{46} + 2(A^2 - I)$
$= A^{44} + 3(A^2 - I)$
$= A^2 + 24(A^2 - I)$
$= 25A^2 - 24I$
$= 25 \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix} - 24 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
$= \begin{bmatrix} 1 & 0 & 0 \\ 25 & 1 & 0 \\ 25 & 0 & 1 \end{bmatrix}$
Sum of all elements $= 1 + 0 + 0 + 25 + 1 + 0 + 25 + 0 + 1 = 53$
Option (1)
Correct Answer: 1