Linear Algebra: Matrices and Linear Equations - Matrices

Questions

Question 1

Show that for any square matrix, the matrix \(A + \mathbf{A}^\top\) is symmetric.

Question 2

If \(A = (a_{ij})\) is a square matrix, then the elements \(a_{ii}\) are called the diagonal elements. How do the diagonal elements of \(A\) and \(\mathbf{A}^\top\) differ?

Question 3

Define a matrix \(A\) to be skew-symmetric if \(\mathbf{A}^\top = -A\). Show that for any square matrix \(A\), the matrix \(A-\mathbf{A}^\top\) is skew-symmetric.

Question 4

If a matrix is skew-symmetric, what can you say about its diagonal elements?

Answers

Answer 1

the matrix \(A + \mathbf{A}^\top\) is symmetric.

\[\because (A + \mathbf{A}^\top)^\top = (\mathbf{A}^\top + A)\] \[= (A+\mathbf{A}^\top)\]

Answer 2

Same

\[A_{i,j} = \begin{pmatrix} \color{green}{a_{1,1}} & a_{1,2} & \cdots & a_{1,j} \\ a_{2,1} & \color{green}{a_{2,2}} & \cdots & a_{2,j} \\ \vdots & \vdots & \ddots & \vdots \\ a_{i,1} & a_{i,2} & \cdots & \color{green}{a_{i,j}} \end{pmatrix}\] \[\mathbf{(A)}^\top_{i,j} = \begin{pmatrix} \color{red}{a_{1,1}} & a_{2,1} & \cdots & a_{i,1} \\ a_{1,2} & \color{red}{a_{2,2}} & \cdots & a_{i,2} \\ \vdots & \vdots & \ddots & \vdots \\ a_{1,j} & a_{2,j} & \cdots & \color{red}{a_{i,j}} \end{pmatrix}\]

Answer 3

The matrix \(A - \mathbf{A}^\top\) is a skew-symmetric.

\[\because \mathbf{(A - \mathbf{A}^\top)}^\top = \mathbf{A}^\top - A\] \[= -(A - \mathbf{A}^\top)\]

Answer 4

\[\text{By skew-symmetric}\] \[a_{ii} = -a_{ii}\] \[\therefore a_{ii} = 0\]