2. Determinants
Determinants
Ex 1.
Calculate the determinant of the matrix
\[
A=\begin{pmatrix}
2 & 3 & 1\\
0 & -1 & 4\\
5 & 2 & 0
\end{pmatrix}
\quad
B=\begin{pmatrix}
1 & 2 & 2\\
4 & 0 & 0\\
7 & 8 & 9
\end{pmatrix}
\qquad
C=\begin{pmatrix}
3 & 0 & 2\\
2 & 0 & -2\\
0 & 1 & 1
\end{pmatrix}
\]
using Sarrus' rule.
Ex 2.
Determine the determinants using Laplace expansion:
\[
A=\begin{pmatrix}
1
\end{pmatrix}
\quad
B=\begin{pmatrix}
1 & 0 & 2\\
3 & 1 & 0\\
4 & 5 & 6
\end{pmatrix}
\quad
C=\begin{pmatrix}
1 & 2 & 3 & 4\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 1\\
0 & 0 & 0 & 2
\end{pmatrix}
\]
Ex 3.
Show that if two rows in a matrix are equal, then the determinant is equal to zero. Give an example of a \(3\times3\) matrix with two equal rows and calculate its determinant. Justify why this happens.
Ex 4.
Calculate the determinant of the triangular matrix \(T\) with diagonal elements \((3,-2,5,1)\).
Ex 5.
For the matrix dependent on parameter \(t\):
\[
M(t)=\begin{pmatrix}
t & 1\\
2 & t\\
\end{pmatrix}
\]
calculate \(\det(M(t))\) and find the values of \(t\) for which the matrix is singular.
Ex 6.
Solve the equation
\[
\det\begin{pmatrix}
x & 3\\
2 & x
\end{pmatrix} = 0
\]
Ex 7.
\(\star\) Solve the equation
\[
\det\begin{pmatrix}
x & 3\\
2 & -x
\end{pmatrix} = 0
\]
Ex 8.
Calculate the determinant of the matrix
\[
\begin{vmatrix}
x & y & x+y \\
y & x+y & x \\
x+y & x & y
\end{vmatrix}
\]
Ex 9.
Show that the equality holds
\[
\begin{vmatrix}
1 & 1 & 1 \\
x & y & z \\
x^2 & y^2 & z^2
\end{vmatrix}
= (z-x)(z-y)(y-x)
\]
Prove a similar equality for the determinant
\[
\begin{vmatrix}
1 & 1 & 1 & 1 \\
x & y & z & u \\
x^2 & y^2 & z^2 & u^2 \\
x^3 & y^3 & z^3 & u^3
\end{vmatrix}
\]
Ex 10.
Calculate the determinant of the matrix
\[
\begin{vmatrix}
a & a & a \\
-a & a & a \\
-a & -a & a
\end{vmatrix}
\quad
\text{\&}
\quad
\begin{vmatrix}
a & 0 & b \\
0 & c & 0 \\
d & 0 & a
\end{vmatrix}
\]
Ex 11.
Check the validity of the following relationships:
a)
\[
\begin{vmatrix}
a+b & b \\
c+d & d
\end{vmatrix}
=
\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}
\]
b)
\[
\begin{vmatrix}
a+bx & b \\
c+dx & d
\end{vmatrix}
=
\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}
\]