If A=[ a 0 0 0 b 0 0 0 c ], then A^10 is (a) 0 (b) (abc)^10

step 1 So in the given question we have a we have a matrix a which is given as a is equal to A 0 0 0 0

If A=[ a 0 0 0 b 0 0 0 ...

If $\mathrm{A}=\left[\begin{array}{lll}\mathrm{a} & 0 & 0 \\ 0 & \mathrm{~b} & 0 \\ 0 & 0 & \mathrm{c}\end{array}\right]$, then $\mathrm{A}^{10}$ is (a) 0 (b) $(\mathrm{abc})^{10}$ (c) $\left[\begin{array}{ccc}a^{10} & 0 & 0 \\ 0 & b^{10} & 0 \\ 0 & 0 & c^{10}\end{array}\right]$ (d) $\left[\begin{array}{ccc}a^{10} & b & c \\ c & b^{10} & a \\ a & b & c^{10}\end{array}\right]$

If $\mathrm{A}=\left[\begin{array}{lll}\mathrm{a} & 0 & 0 \\ 0 & \mathrm{~b} & 0 \\ 0 & 0 & \mathrm{c}\end{array}\right]$, then $\mathrm{A}^{10}$ is
(a) 0
(b) $(\mathrm{abc})^{10}$
(c) $\left[\begin{array}{ccc}a^{10} & 0 & 0 \\ 0 & b^{10} & 0 \\ 0 & 0 & c^{10}\end{array}\right]$
(d) $\left[\begin{array}{ccc}a^{10} & b & c \\ c & b^{10} & a \\ a & b & c^{10}\end{array}\right]$

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Key Concepts

Diagonal Matrix

A diagonal matrix is a type of square matrix in which the entries outside the main diagonal are all zero. This property simplifies many operations in linear algebra, such as finding powers of the matrix, inverses (when they exist), and determinant calculations, as the calculations reduce to operating only on the diagonal elements.

Matrix Exponentiation

Matrix exponentiation refers to raising a matrix to a given power, typically by repeated matrix multiplication. For a diagonal matrix, this process is simplified because the power of the matrix is obtained by simply raising each of its diagonal elements to the same power, while still keeping the off-diagonal entries zero.

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