# Matrices and Determinants: Problems with Solutions

Problem 1
What are the dimensions of the matrix $A$?
$A=\left[ \begin{array}{ccccc} 2 & -2 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 & 3 \\ 1 & -1 & 3 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 \end{array} \right]$
Problem 2
$A=\left[ \begin{array}{ccccc} 2 & -2 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 & 3 \\ 1 & -1 & 3 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1% \end{array}% \right]$

Which is the element $A_{2,4}$?
Problem 3
$A=\left[ \begin{array}{ccccc} 2 & -2 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 & 3 \\ 1 & -1 & 3 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1% \end{array}% \right]$

Which is the element $A_{3,2}$?
Problem 4 sent by Oyoo Fredrick Ochieng
Define the term identity matrix (unit matrix).
Problem 5
Write the following system of equations as an augmented matrix.

$\left\{ \begin{array}{c} 3x-2y=3 \\ 5x+y=0 \end{array} \right\}$
Problem 6
What is the sum of the matrices?
$\left[ \begin{array}{cc} 2 & -1 \\ 1 & 3 \end{array} \right] +\left[ \begin{array}{cc} 1 & 0 \\ 2 & -1 \end{array} \right] =$
Problem 7
Find the matrix $A$, so that the following equality is satisfied.

$A+\left[ \begin{array}{cc} 2 & 3 \\ -4 & 1 \end{array} \right] =\left[ \begin{array}{cc} 5 & -1 \\ 1 & 5 \end{array} \right]$
Problem 8
What is the result of the multiplication?
$5 \times \left[ \begin{array}{c} -2 \\ 3 \\ -4 \end{array} \right] =$
Problem 9
Find the matrix $X$.

$\frac{3}{2}X+\left[ \begin{array}{cc} -1 & 3 \\ 2 & -2 \end{array} \right] =\left[ \begin{array}{cc} 3 & -4 \\ 5 & 4 \end{array} \right]$
Problem 10
If $A=B\times C$, find the matrix $A$.

$B=\left[ \begin{array}{ccc} 1 & -3 & -2 \\ 2 & 0 & 1 \end{array} \right]$      $C=\left[ \begin{array}{cc} 2 & 1 \\ -2 & -1 \\ 3 & 0 \end{array} \right]$

Problem 11
Find the determinant of the matrix.
$A=\left[ \begin{array}{cc} 2 & -3 \\ 4 & 5 \end{array} \right]$
Problem 12
Find the determinant of the matrix.
$A=\left[ \begin{array}{cc} 3 & 4 \\ 0 & 0 \end{array} \right]$
Problem 13
Find the inverse of the matrix $A=\left[ \begin{array}{cc} 2 & -3 \\ 4 & 5 \end{array} \right]$
Problem 14
Find the inverse of the matrix $A=\left[ \begin{array}{cc} 0 & \frac{-3}{4} \\ \frac{7}{3} & 0 \end{array} \right]$
Problem 15
Find the inverse of the matrix $A=\left[ \begin{array}{cc} 3 & -4 \\ -6 & 8 \end{array} \right]$
Problem 16
$A=\left[ \begin{array}{cc} 7 & -4 \\ 4 & -3 \end{array} \right]$ $B=\left[ \begin{array}{cc} \frac{3}{5} & -\frac{4}{5} \\ \frac{4}{5} & -\frac{7}{5} \end{array} \right]$
Are $A$ and $B$ multiplicative inverse (can we write that $A \cdot B = B \cdot A$)?
Problem 17
$A=\left[ \begin{array}{cc} 2 & -3 \\ 1 & -2 \end{array} \right]$   $B=\left[ \begin{array}{cc} -2 & 1 \\ -3 & 2% \end{array} \right]$
Are $A$ and $B$ multiplicative inverse (Can we write that $A \cdot B = B \cdot A$)?
Problem 18
$A=\left[ \begin{array}{cc} 8 & 9 \\ -1 & 2 \end{array} \right]$   $B=\left[ \begin{array}{cc} \frac{2}{25} & -\frac{1}{5} \\ \frac{3}{25} & \frac{9}{25} \end{array} \right]$
Are $A$ and $B$ multiplicative inverse?
Problem 19
$A=\left[ \begin{array}{cc} 8 & 9 \\ -1 & 2 \end{array} \right]$   $B=\left[ \begin{array}{cc} \frac{2}{25} & -\frac{9}{25} \\ \frac{1}{25} & \frac{8}{25} \end{array} \right]$
Are $A$ and $B$ multiplicative inverse?
Problem 20
What value must $x$ have, so that $B$ is the inverse of $A$?
$A=\left[ \begin{array}{cc} 1 & 3 \\ -1 & 2 \end{array}% \right] \qquad B=\left[ \begin{array}{cc} \frac{2}{5} & x \\ \frac{1}{5} & \frac{1}{5} \end{array} \right]$
Problem 21
What is the value of $x$, so the matrix $A$ does not have an inverse?

$A=\left[ \begin{array}{cc} 2 & 3 \\ x & -2 \end{array} \right]$
Problem 22
What value must $x$ have, so the matrix $A$ does not have an inverse?

$A=\left[ \begin{array}{cc} 1 & 2+x \\ x & -1 \end{array} \right]$

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