For a 3x3 matrix, find the determinant by first
Another way to think of transposing is that you rewrite the first row as the first column, the middle row becomes the middle column, and the third row becomes the third column. Notice the colored elements in the diagram above and see where the numbers have changed position.
In the example shown above, if you want the minor matrix of the term in the second row, first column, you highlight the five terms that are in the second row and the first column. The remaining four terms are the corresponding minor matrix. Find the determinant of each minor matrix by cross-multiplying the diagonals and subtracting, as shown. For more on minor matrices and their uses, see Understand the Basics of Matrices.
When assigning signs, the first element of the first row keeps its original sign. The second element is reversed. The third element keeps its original sign. Continue on with the rest of the matrix in this fashion. Note that the (+) or (-) signs in the checkerboard diagram do not suggest that the final term should be positive or negative. They are indicators of keeping (+) or reversing (-) whatever sign the number originally had. The final result of this step is called the adjugate matrix of the original. This is sometimes referred to as the adjoint matrix. The adjugate matrix is noted as Adj(M).
For the sample matrix shown in the diagram, the determinant is 1. Therefore, dividing every term of the adjugate matrix results in the adjugate matrix itself. (You won’t always be so lucky. ) Instead of dividing, some sources represent this step as multiplying each term of M by 1/det(M). Mathematically, these are equivalent.
Recall that the identity matrix is a special matrix with 1s in each position of the main diagonal from upper left to lower right, and 0s in all other positions.
Remember that row reductions are performed as a combination of scalar multiplication and row addition or subtraction, in order to isolate individual terms of the matrix. For a more complete review, see Row-Reduce Matrices.
If you wish to enter a negative number, use your calculator’s negative button (-) and not the minus key. The matrix function will not read the number properly. If necessary, you can use your calculator’s arrow keys to jump around the matrix.
Do not use the ^ button on your calculator to try entering A^-1 as separate keystrokes. The calculator will not understand this operation. If you receive an error message when you enter the inverse key, chances are that your original matrix does not have an inverse. You may want to go back and calculate the determinant to find out.
Your calculator probably has a function that will automatically convert the decimals to fractions. For example, using the TI-86, enter the Math function, then select Misc, and then Frac, and Enter. The decimals will automatically appear as fractions.
