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: \(A=\begin{bmatrix}2 & 0 & 1 \\0 & -1 & 1 \\3 & 1 & -2\end{bmatrix}\)

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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.

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

• \(\color{blue}{27}\)
• \(\color{blue}{-4}\)
• \(\color{blue}{23}\)
• \(\color{blue}{-13}\)