Question

Question: If the$\mathbf{n}\mathbf{\times }\mathbf{n}$matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?$\mathbf{2}{\mathbf{l}}_{\mathbf{n}}\mathbf{+}\mathbf{3}\mathbf{A}\mathbf{-}\mathbf{4}{\mathbf{A}}^{\mathbf{2}}$.

Short answer

The Matrix$2{\mathrm{l}}_{\mathrm{n}}+3\mathrm{A}-4{\mathrm{A}}^2$ is symmetric.

Step-by-step solution

Definition of symmetric matrix.

A square matrix is symmetric matrix if$\mathbf{A}\mathbf{=}{\mathbf{A}}^{\mathbf{T}}$

Verification whether the given matrix is symmetric.

Given that A and B are symmetric matrices and B is invertible.

Since A is symmetric, then $A=A^T$.

Then by properties of transpose,${\left(A+B\right)}^T=A^T+B^T$ it gives,

$$\begin{gathered} {\left(2{\mathrm{l}}_{\mathrm{n}}+3\mathrm{A}-4{\mathrm{A}}^2\right)}^T={\left(2{\mathrm{l}}_{\mathrm{n}}\right)}^T+{\left(3\mathrm{A}\right)}^T-{\left(4{\mathrm{A}}^2\right)}^T \\ =2{\mathrm{l}}_{\mathrm{n}}+3{\mathrm{A}}^{\mathrm{T}}-{\left(4{\mathrm{A}}^{\mathrm{T}}\right)}^2 \\ =2{\mathrm{l}}_{\mathrm{n}}+3\mathrm{A}-4{\mathrm{A}}^2 \end{gathered}$$

Therefore, $2{\mathrm{l}}_{\mathrm{n}}+3\mathrm{A}-4{\mathrm{A}}^2$is a symmetric matrix.