Learn Matrices: Practice Problems and Examples

Prove that if B is symmetric, then $\Vert B\Vert$ is the largest eigenvalue of B.

Describe all skew symmetric scalar matrices.

$\left[\begin{array}{ccc}0& 2& 4\\ 4& 4& 1\end{array}\right]-\left[\begin{array}{ccc}2& 3& 3\\ -2& 4& -2\end{array}\right]$

Prove that the eigenvalues of a Hermitian matrix are real.

Problem 3. Find the unitary operator $\hat{U}$ which diagonalizes one of the Pauli matrices
${\hat{\sigma }}_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$
Obtain the eigenvalues and eigenvectors of ${\hat{\sigma }}_x$

If A is diagonalizable and for all eigenvalues , $\lambda \text{ of }A,|\lambda |=1$ , then A is unitary. True or False?

In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.

Describe how to subtract matrices.

Matrix multiplication is pretty tough- so i will cover that in class. In the meantime , compute the following if
$A=\left[\begin{array}{ccc}2& 1& 1\\ -1& -1& 4\end{array}\right],B=\left[\begin{array}{cc}0& 2\\ -4& 1\\ 2& -3\end{array}\right],C=\left[\begin{array}{cc}6& -1\\ 3& 0\\ -2& 5\end{array}\right],D=\left[\begin{array}{ccc}2& -3& 4\\ -3& 1& -2\end{array}\right]$
If the operation is not possible , write NOT POSSIBLE and be able to explain why
a)A+B
b)B+C
c)2A

Suppose that A and B are diagonalizable matrices. Prove or disprove that A is similar to B if and only if A and B are unitarily equivalent.

Find if possible the matrices:
a) AB
b) BA
$A=\left[\begin{array}{c}-1\\ -2\\ -3\end{array}\right],B=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$

Find the products AB and BA for the diagonal matrices.
$A=\left[\begin{array}{ccc}3& 0& 0\\ 0& -5& 0\\ 0& 0& 0\end{array}\right],B=\left[\begin{array}{ccc}-7& 0& 0\\ 0& 4& 0\\ 0& 0& 12\end{array}\right]$

Simplify the following matrices , then find the multiplication AB
$A=\left[\begin{array}{cc}4i& \sqrt{-1}\\ \sqrt{2}{e}^{i\pi /4}& 5\sin (53.13)\\ \sqrt{-9}& \sqrt{1}\end{array}\right]$ and $B=\left[\begin{array}{ccc}\ln e^3& {\log }_x{x}^2& \sqrt{-1}\ln e^i\\ i{\log }_y{y}^{2i}& 2e^{i\pi }& \ln e^i\end{array}\right]$

Given the matrices
$A=\left[\begin{array}{ccc}1& -1& 2\\ 3& 4& 5\\ 0& 1& -1\end{array}\right],B=\left[\begin{array}{ccc}0& 2& 1\\ 3& 0& 5\\ 7& -6& 0\end{array}\right]\text{ and }C=\left[\begin{array}{ccc}0& 0& 2\\ 3& 1& 0\\ 0& -2& 4\end{array}\right]$
Determine the following
i)2A-B+2C ii)A+B+C iii)4C-2B+3A iv)$(A\times B)-C$

Find x and y.
$\left[\begin{array}{cc}x& -2\\ 7& y\end{array}\right]=\left[\begin{array}{cc}-4& -2\\ 7& 22\end{array}\right]$

Find AB, given shown matrices,
If $A=\left[\begin{array}{ccc}2& 0& 4\end{array}\right]\text{ and }B=\left[\begin{array}{c}1\\ 3\\ 7\end{array}\right]$

If A,B are symmetric matrices, then prove
that $(B{A}^{-1}{)}^T(A^{-1}{B}^T{)}^{-1}=I$

Let $A=\left[\begin{array}{cc}1& 2\\ 4& -1\end{array}\right]\text{ and }B=\left[\begin{array}{cc}a& -3\\ -6& 1\end{array}\right]$ . For what values of a (if any) do matrices A and B commute?
None
-2
-6
-1
-4

Define first the scalar product of a row vector a and a column vector b, each having the same number p of elements?
$a=[a_1{a}_2\dots a_p]\text{ and }b={[b_1{b}_2\dots b_p]}^T$