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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 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 = 
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 = 
then the determinant |A| = 
.
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.
