Matrix Determinant Help
The last computation we will learn is finding a matrix’s determinant . Although we will not use the determinant here, it is used in vector mathematics courses, some theoretical algebra courses, and in algebra courses that cover Cramer’s Rule (used to solve systems of linear equations). An interesting fact about determinants is that a square matrix has an inverse only when its determinant is a nonzero number.
The usual notation for a determinant is to enclose the matrix using two vertical bars instead of two brackets. The determinant for the matrix .
Finding the determinant for a 2 × 2 matrix is not hard.
We find the determinant of larger matrices by breaking down the larger matrix into several 2 × 2 sub-matrices. For larger matrices, there are numerous formulas for computing their determinants. Some of them come from expanding the matrix along each row and along each column. This means that we will multiply the entries in a row or a column by the determinant of a smaller matrix. This smaller matrix comes from deleting the row and column an entry is in. When working with a 3 × 3 matrix, these sub-matrices will be 2 × 2 matrices.
Suppose we want to expand the following matrix along the first row.
We will multiply the A entry by the submatrix obtained by removing the first row ABC and the first column . This leaves us with the matrix . Our first calculation will be
Similarly, when we use entry B, we will need to remove the first row A B C and the second column . This leaves us with . There is a complication—the signs on the entries must alternate when we perform these expansions. For our matrix, the signs will alternate beginning with A not changing, but B and D changing.
For our 3 × 3 matrix, the expansion along the first row looks like this.
The expansion along the second column looks like this.
- Find the determinant for
We will use two calculations, along Row 2 and along Column 3. By Row 2 we have
By Column 3 we have
The method is the same for larger matrices except that there are more levels of work.
Expanding this matrix along Row 1 gives us
Each of these four determinants must be computed using the previous method for a 3 × 3 matrix.
1. Expanding this matrix along Row 2, we have
Practice problems for this concept can be found at: Matrices Practice Test
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