How to Find the Determinant of a 3x3 Matrix by Hand

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The determinant is a special real number associated with matrices that is useful in linear equations and calculus. Finding the determinant of a three-by-three matrix involves a long, complicated formula that is derived from a pattern of multiplication and addition using the numbers in the matrix. When determining the matrix by hand, however, you can use a shortcut method to quickly find the answer without delving into the formula.

1 Repeat the first two columns

Repeat the first two columns to the right of the third column. As an example, if the three-by-three matrix consecutively used the numbers one through nine, the first row would then read "1 2 3 1 2." The second row would read "4 5 6 4 5." The last row would read "7 8 9 7 8."

2 Multiply the numbers

Multiply the numbers diagonally down from left to right. Only multiply diagonals that contain three numbers. In the example, multiply 1 by 5 by 9 on the first diagonal to get 45. Multiply 2 by 6 by 7 on the second diagonal to get 84. Multiply 3 by 4 by 8 on the third diagonal to get 96.

3 Add the three products together

Add the three products together. In the example, 45 plus 84 plus 96 results in a total of 225.

4 Multiply the numbers-2

Multiply the numbers diagonally up from left to right. Only multiple diagonals that contain three numbers. Continuing with the example, multiply 7 by 5 by 3 on the first diagonal to get 105. Multiply 8 by 6 by 1 on the second diagonal to get 48. Multiply 9 by 4 by 2 on the third diagonal to get 72.

5 Add the three products

Add the three products together. In the example, 105 plus 48 plus 72 gives you 225.

6 Subtract the second total from the first total

Subtract the second total from the first total to find the determinant. In the example, 225 minus 225 calculates a determinant of zero.

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