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.
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."
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.
Add the three products together. In the example, 45 plus 84 plus 96 results in a total of 225.
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.
Add the three products together. In the example, 105 plus 48 plus 72 gives you 225.
Subtract the second total from the first total to find the determinant. In the example, 225 minus 225 calculates a determinant of zero.