Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
Let $A = \begin{bmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{bmatrix}$. If $A^2 = \begin{bmatrix} d_1^2 & 0 & 0 \\ 0 & d_2^2 & 0 \\ 0 & 0 & d_3^2 \end{bmatrix}$ and $kA = \begin{bmatrix} kd_1 & 0 & 0 \\ 0 & kd_2 & 0 \\ 0 & 0 & kd_3 \end{bmatrix}$, then find the total number of possible $3 \times 3$ matrices where each diagonal element is from $\{0, 1, -1\}$.
Step-by-Step Solution
Key Concept: For a diagonal matrix with 3 independent diagonal elements each having 3 possible values, the total count is $3^3$, minus the null matrix if required.
A diagonal matrix has the form $A = \begin{bmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{bmatrix}$ where each $d_i \in \{0, 1, -1\}$. Since each of the three diagonal positions can independently take any of 3 values, the total number of possible matrices is $3^3 = 27$. However, the problem states "$A$ can't be a null matrix," so we exclude the case where $d_1 = d_2 = d_3 = 0$. Therefore, the total number of possible matrices is $27 - 1 = 26$.
Correct Answer: 26