If A and B are diagonal matrices of order n with diagonal elements a1, a2, …an and b1, b2, …, b

step 1 In the problem late, A equal A100, 0, A2, 0, dot, dot, dot, dot, dot, dot. Lastly, we have AN. A

If A and B are diagonal matrices of order n with diagonal...

If $\mathrm{A}$ and $\mathrm{B}$ are diagonal matrices of order $\mathrm{n}$ with diagonal elements $\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots \mathrm{a}_{\mathrm{n}}$ and $\mathrm{b}_{1}, \mathrm{~b}_{2}, \ldots, \mathrm{b}_{\mathrm{n}}$, then for $\mathrm{k} \geq 1,(\mathrm{~A}+\mathrm{B})^{\mathrm{k}}$ is a (a) diagonal matrix with diagonal elements $\left(a_{1}+b_{1}\right)^{k},\left(a_{2}+b_{2}\right)^{k}, \ldots . .\left(a_{n}+b_{n}\right)^{k}$ (b) diagonal matrix with diagonal elements $\left(a_{1} b_{1}\right)^{k},\left(a_{2} b_{2}\right)^{k}, \ldots \ldots\left(a_{n} b_{n}\right)^{k}$ (c) diagonal matrix with diagonal elements $\left(a_{1}^{k}+b_{1}^{k}\right), \ldots . .\left(a_{n}^{k}+b_{n}^{k}\right)$ (d) None of these

If $\mathrm{A}$ and $\mathrm{B}$ are diagonal matrices of order $\mathrm{n}$ with diagonal elements $\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots \mathrm{a}_{\mathrm{n}}$ and $\mathrm{b}_{1}, \mathrm{~b}_{2}, \ldots, \mathrm{b}_{\mathrm{n}}$, then for $\mathrm{k} \geq 1,(\mathrm{~A}+\mathrm{B})^{\mathrm{k}}$
is a
(a) diagonal matrix with diagonal elements $\left(a_{1}+b_{1}\right)^{k},\left(a_{2}+b_{2}\right)^{k}, \ldots . .\left(a_{n}+b_{n}\right)^{k}$
(b) diagonal matrix with diagonal elements $\left(a_{1} b_{1}\right)^{k},\left(a_{2} b_{2}\right)^{k}, \ldots \ldots\left(a_{n} b_{n}\right)^{k}$
(c) diagonal matrix with diagonal elements $\left(a_{1}^{k}+b_{1}^{k}\right), \ldots . .\left(a_{n}^{k}+b_{n}^{k}\right)$
(d) None of these

Key Concepts

Matrix Addition

Matrix addition is performed by adding corresponding entries of the matrices involved. For diagonal matrices, since off-diagonal entries are zero, the addition simply sums the diagonal elements independently. This means that when two diagonal matrices are added, the resulting matrix is also diagonal with each diagonal element being the sum of the diagonal elements of the original matrices.

Matrix Exponentiation

Matrix exponentiation involves raising a matrix to a given power, which for diagonal matrices is very straightforward. With a diagonal matrix, raising it to a power means raising each of its diagonal entries to that power. Moreover, when dealing with the sum of two diagonal matrices, and then raising the result to an exponent, the operation can be handled elementwise. This is why taking powers, after performing an elementwise operation like addition in the context of diagonal matrices, results in a diagonal matrix where each diagonal element is the corresponding summed and then exponentiated element.

Diagonal Matrices

A diagonal matrix is a square matrix in which all off-diagonal entries are zero. This property simplifies many matrix operations because many functions, such as addition, multiplication, and taking powers, act elementwise on the diagonal entries. The structure of diagonal matrices allows us to perform operations independently on each diagonal element, thereby reducing complex matrix computations to simpler scalar operations.

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