How to Find Determinants of a Matrix?

A Matrix is an array of numbers: $m×n$ with $m$ rows and $n$ columns. The determinant of a matrix is a scalar value that is defined for square matrices.

Related Topics

• How to Multiply a Matrix by a Scalar
• How to Add and Subtract Matrices

A step-by-step guide to finding determinants of a matrix

• The determinant of a $2×2$ matrix $A$, $A=\begin{bmatrix}a & b \\c & d \end{bmatrix}$ is defined as $|A|=ad-bc$.
• The determinant of a $3×3$ matrix $A$, $A=\begin{bmatrix}a & b & c \\d & e & f \\g & h & i\end{bmatrix}$ is defined as $|A|=a(ei-fh)-b(di-fg)+c(dh-eg)$

Finding Determinants of a Matrix – Example 1:

Evaluate the determinant of matrix $A=\begin{bmatrix}5 & -1 \\6 & 2 \end{bmatrix}$

Solution:

The determinant is: $|A|=5(2)-(-1)(6)=10-(-6)=10+6=16$

Finding Determinants of a Matrix – Example 2:

Evaluate the determinant of matrix: $A=\begin{bmatrix}2 & 0 & 1 \\0 & -1 & 1 \\3 & 1 & -2\end{bmatrix}$

Solution:

The determinant is: $|A|=2((-1)(-2)-(1)(1))-0((-2)(0)-(1)(3)+1((0)(1)-(-1)(3))=5$

Exercises for Finding Determinants of a matrix

Evaluate the determinants of each matrix.

• $\color{blue}{\begin{bmatrix}3 & 5 \\0 & 9 \end{bmatrix}}$
• $\color{blue}{\begin{bmatrix}0 & 1 \\4 & 6 \end{bmatrix}}$
• $\color{blue}{\begin{bmatrix}1 & 5 & 4\\0 & 9 & 1\\1 & 0 & 6\end{bmatrix}}$
• $\color{blue}{\begin{bmatrix}6 & 0 & 5\\1 & 4 & 2\\3 & 7 & 4\end{bmatrix}}$

• $\color{blue}{27}$
• $\color{blue}{-4}$
• $\color{blue}{23}$
• $\color{blue}{-13}$