<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation
Our Terms of Use (click here to view) and Privacy Policy (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use and Privacy Policy.

Determinants

Number calculated from the entries in a matrix primarily through multiplication and subtraction.

Atoms Practice
Estimated18 minsto complete
%
Progress
Practice Determinants
Practice
Progress
Estimated18 minsto complete
%
Practice Now
Finding Determinants of Matrices

A mathematical theorem states that a matrix is singular if and only if its determinant is zero. Is the following matrix singular?

\begin{align*}\begin{bmatrix} 2 & 1 & 3\\ 0 & 2 & 1\\ -1 & 3 & 0\end{bmatrix}\end{align*}

Determinant of a Matrix

Each square matrix has a real number value associated with it called its determinant. This value is denoted by \begin{align*}det \ A\end{align*} or \begin{align*}|A|\end{align*}.

Finding the Determinant of a \begin{align*}2 \times 2\end{align*} matrix:

\begin{align*}det\begin{bmatrix} a & b\\ c & d\end{bmatrix} = \begin{vmatrix} {\color{red}a} & {\color{blue}b}\\ {\color{blue}c} & {\color{red}d}\end{vmatrix} = {\color{red}ad}- {\color{blue}bc}\end{align*}

Finding the Determinant of a \begin{align*}3 \times 3\end{align*} matrix: To begin, we will repeat the first two columns after matrix. Next, calculate the products and sums as shown below and find the difference. The result is the determinant of the \begin{align*}3 \times 3\end{align*} matrix.

Using the determinant to find the Area of a Triangle in the coordinate plane:

We can find the area of a triangle with vertices \begin{align*}(x_1,y_1), (x_2,y_2)\end{align*} and \begin{align*}(x_3,y_3)\end{align*} using the formula below

\begin{align*}A = \pm\frac{1}{2}\begin{vmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ x_3 & y_3 & 1\end{vmatrix},\end{align*} where the \begin{align*}\pm\end{align*} accounts for the possibility that the determinant could be negative but area should always be positive.

Using the calculator to find the determinant of a matrix: If you are using a TI-83 or TI-84, access the Matrix menu by either pressing MATRIX or (\begin{align*}2^{nd} \ x^{-1}\end{align*} MATRIX). Now you can choose to EDIT matrix \begin{align*}A\end{align*}. Change the dimensions as needed and enter the data values. Now return to the home screen (\begin{align*}2^{nd}\end{align*} MODE QUIT) and return to the MATRIX menu. Arrow over to MATH and select 1:det( by pressing ENTER. Go into the MATRIX menu once more to select 1:\begin{align*}[A]\end{align*} under the NAMES column. Press ENTER. Your screen should show \begin{align*}det([A]\end{align*} at this time. Press ENTER once more and the result will be your determinant. These directions work for square matrices of any size.

Solve the following problems

Find the \begin{align*}det\begin{bmatrix} 3 & -4\\ 1 & 5\end{bmatrix}.\end{align*}

Using the rule above for a \begin{align*}2 \times 2\end{align*} matrix, the determinant can be found as shown:

\begin{align*}det\begin{bmatrix} 3 & -4\\ 1 & 5\end{bmatrix} = \begin{vmatrix} {\color{red}3} & {\color{blue}-4}\\ {\color{blue}1} & {\color{red}5}\end{vmatrix} = {\color{red}(3)(5)}-{\color{blue}(-4)(1)} = {\color{red}15}-{\color{blue}(-4)} = 19\end{align*}

Find the \begin{align*}det\begin{bmatrix}2 & -3 & 5\\ -4 & 7 & 1\\ 3 & 8 & 6\end{bmatrix}.\end{align*}

First, we need to repeat the first two columns. Then we can find the diagonal products as shown:

\begin{align*}\begin{bmatrix} 2 & -3 & 5\\ -4 & 7 & 1\\ 3 & 8 & 6\end{bmatrix}\begin{matrix} 2 & -3\\ -4 & 7\\ 3 & 8\end{matrix}\\ {\color{red}(2 \cdot 7 \cdot 6)+(-3 \cdot 1 \cdot 3)+(5\cdot-4\cdot8)} & \ {\color{red}= 84+-9+-160 = -85}\\ = \ {\color{blue}(3\cdot7\cdot5)+(8\cdot 1\cdot 2)+(6\cdot-4\cdot-3)} & \ {\color{blue}= 105+16+72 = 193}\\ {\color{red}-85}-{\color{blue}193} &= -278\end{align*}

Find the area of the triangle with vertices (2, -1), (4, 5) and (8, 1)

The first step is to set up our matrix and find the determinant as shown:

\begin{align*}\begin{bmatrix} 2 & -1 & 1\\ 4 & 5 & 1\\ 8 & 1 & 1\end{bmatrix}\begin{matrix} 2 & -1\\ 4 & 5\\ 8 & 1\end{matrix}\\ {\color{red}(2\cdot5\cdot1)+(-1\cdot1\cdot8)+(1\cdot4\cdot1)} & \ {\color{red}=10+-8+4=6}\\ = \ {\color{blue}(8\cdot5\cdot1)+(1\cdot1\cdot2)+(1\cdot4\cdot-1)} & \ {\color{blue}= 40+2-4=38}\\ {\color{red}6}-{\color{blue}38} &= -32\end{align*}

Now we can multiply this determinant, -32, by \begin{align*}-\frac{1}{2}\end{align*} (we will multiply by the negative in order to have a positive result) to get 16. So the area of the triangle is 16 \begin{align*}u^{2}\end{align*}.

Examples

Example 1

Earlier, you were asked if the matrix is singular. 

To find the determinant, we first need to repeat the first two columns. Then we can find the diagonal products as shown:

\begin{align*}\begin{bmatrix} 2 & 1 & 3\\ 0 & 2 & 1\\ -1 & 3 & 0\end{bmatrix}\begin{matrix} 2 & 1\\ 0 & 2\\ -1 & 3\end{matrix}\\ {\color{red}(2 \cdot 2 \cdot 0)+(1 \cdot 1 \cdot -1)+(3 \cdot 0 \cdot 3)} & \ {\color{red}= 0+ -1 + 0 = -1}\\ = \ {\color{blue}(3 \cdot 2 \cdot -1)+(2 \cdot 1 \cdot 3)+(1 \cdot 0 \cdot 0)} & \ {\color{blue}= -6 + 6 + 0 = 0}\\ {\color{red}-1}-{\color{blue}0} &= -1\end{align*}

The determinant is not zero and therefore the matrix is not singular.

Find the determinants of the matrices below.

Example 2

\begin{align*}\begin{bmatrix} -1 & 8\\ 2 & -9\end{bmatrix}\end{align*}


\begin{align*}\begin{vmatrix} {\color{red}-1} & {\color{blue}8}\\ {\color{blue}2} & {\color{red}-9}\end{vmatrix} = {\color{red}(-1)(-9)}-{\color{blue}(8)(2)} = {\color{red}9}-{\color{blue}16} = -7\end{align*}

Example 3
\begin{align*}\begin{bmatrix} -2 & 4 & -3\\ 5 & -6 & 1\\ -4 & 1 & -2\end{bmatrix}\end{align*}


\begin{align*}\begin{vmatrix} -2 & 4 & -3\\ 5 & -6 & 1\\ -4 & 1 & -2\end{vmatrix}\begin{matrix}-2 & 4\\ 5 & -6\\ -4 & 1\end{matrix}\\ {\color{red}(-2\cdot-6\cdot-2)+(4\cdot-1\cdot-4)+(-3\cdot5\cdot1)} & \ {\color{red}= -24+16-15=-23}\\ = \ {\color{blue}(-4\cdot-6\cdot-3)+(1\cdot1\cdot-2)+(-2\cdot5\cdot4)} & \ {\color{blue}= -72-2-40=-114}\\ {\color{red}-23}-{\color{blue}(-114)} &= 91\end{align*}

Example 4

Find the area of the triangle with vertices (-5, 2), (8, -1) and (3, 9)

\begin{align*}\begin{vmatrix} -5 & 2 & 1\\ 8 & -1 & 1\\ 3 & 9 & 1\end{vmatrix}\begin{matrix} -5 & 2\\ 8 & -1\\ 3 & 9\end{matrix}\\= \ {\color{red}(5+6+72)}-{\color{blue}(-3 + -45 + 16)} = {\color{red}83}-{\color{blue}(-32)} = 115\end{align*}

So the area is \begin{align*}\frac{1}{2}(115) = 57.5 \ u^{2}\end{align*}.

Review

Find the determinants of the matrices. Use your calculator to check your answers.

  1. .
\begin{align*}\begin{bmatrix} 2 & -1\\ 3 & 5\end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} -3 & -2\\ 6 & 4\end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} 5 & 10\\ -3 & -7\end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} -4 & 8\\ 3 & 5\end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} 11 & 3\\ 7 & 2\end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} 9 & 3\\ 2 & -1\end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} 1 & -1 & 3\\ 5 & 0 & 6\\ -4 & 8 & 2\end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} 5 & -2 & 1\\ 6 & 1 & 0\\ -3 & 2 & 4\end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} 4 & -1 & 2\\ 3 & 0 & 1\\ -2 & 5 & 6\end{bmatrix}\end{align*}

Find the area of each triangle with vertices given below.

  1. (2, -1), (-5, 2) and (0, 6)
  2. (-8, 12), (10, 5) and (1, -4)
  3. (-7, 2), (8, 0) and (3, -4)

Find the value of \begin{align*}a\end{align*} in the matrices below.

  1. .
\begin{align*}\begin{vmatrix} a & 3\\ 8 & 2\end{vmatrix} = -10\end{align*}
  1. .
\begin{align*}\begin{vmatrix} 4 & a\\ 3 & 5\end{vmatrix} = -1\end{align*}
  1. .
\begin{align*}\begin{vmatrix} 2 & -1 & 3\\ 4 & 5 & 2\\ -3 & 0 & a\end{vmatrix} = 23\end{align*}

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 4.7. 

My Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / notes
Show More

Vocabulary

determinant

The determinant is a single number descriptor of a square matrix. The determinant is computed from the entries of the matrix, and has many properties and interpretations explored in linear algebra.

Sarrus’ rule

Sarrus’ rule is a memorization technique that enables you to compute the determinant of matrices efficiently.

Square matrix

A square matrix is a matrix in which the number of rows equals the number of columns.

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Determinants.
Please wait...
Please wait...