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Determinant of a Matrix

Lesson Contents

What is the Matrix Determinant?

The determinant is a number we can calculate from a square matrix. It describes certain properties of the matrix and can be used for solving linear systems of equations. The determinant of a matrix is notated with vertical bars similar to absolute value notation. For example, the determinant of a matrix A is notated as |A|.

The method for finding the determinant depends on the size of the matrix. In this lesson, we will show how to find the determinant of 1×1, 2×2, and 3×3 matrices. A 4×4 or larger matrix takes very long to calculate by hand and is very tedious. For matrices of that size, we recommend using a calculator such as the Determinant Calculator.

how to solve a 3x3 matrix determinant
Example of a shortcut method for the determinant of a 3×3 matrix.

How to Find the Determinant of a 1x1 Matrix

The determinant of a 1×1 matrix is the number inside the matrix.
For example, if matrix A = [5] then the determinant is |A| = 5.

How to Find the Determinant of a 2x2 Matrix

For a 2×2 matrix, if matrix A = 2x2 matrix then the determinant is |A| = ad – bc.

An easy way to remember the order of components that get multiplied and subtracted is to imagine the diagonal line and direction that the components are on.

The 2×2 determinant goes top left to bottom right minus top right to bottom left. That order of diagonal and direction gives us ad – bc.

How to Find the Determinant of a 3x3 Matrix

If matrix A = 3x3 matrix then the determinant |A| = 3x3 matrix determinant.

The top row components of the 3×3 matrix are always the leading coefficients on each smaller 2×2 determinant within the formula. Make sure to remember that there is a negative sign in front of b!

A simple way to remember the 3×3 matrix determinant is to imagine it as three 2×2 matrix determinants. The top row components are leading coefficients in front of each 2×2 determinant.

Break the 3×3 matrix down into the top row 1×3 matrix, and the bottom two rows as a 2×3 matrix. The 2×3 matrix has three combinations of 2×2 matrices in it. Simply memorize the pattern of 1×3 components that multiply by the three 2×2 matrix determinants.

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