There are other important concepts related to matrices. These include the determinant, cofactor, and adjoint of a matrix.
A determinant is a value associated with a square matrix. As a mathematical tool, determinants are of value in helping to solve a series of simultaneous equations.
A 2-row by 2-column determinant can be expressed by enclosing vertical lines around the matrix, as shown here:
Similarly, a determinant is indicated as
One common procedure for finding the determinant of a or matrix is to draw its primary and secondary diagonals. In the case of a determinant, the value is found by multiplying the numbers on the primary diagonal and subtracting from that product the product of the numbers on the secondary diagonal:
For a matrix, we redraw the first two columns to help visualize all diagonals and follow a similar procedure:
Let’s use this approach to find the numerical values of the following and determinants:
A set of simultaneous equations can be solved through the use of determinants by setting up a ratio of two special determinants for each unknown variable. This fairly easy procedure is best illustrated with an example.
Given the three simultaneous equations
we can structure determinants to help solve for unknown quantities and Z:
Determining the values of X, Y, and Z now involves finding the numerical values of the four separate determinants using the method shown earlier in this module:
To verify that and we may choose any one of the original three simultaneous equations and insert these numbers. For example,
Two more useful concepts in the mathematics of matrices are the matrix of cofactors and the adjoint of a matrix. A cofactor is defined as the set of numbers that remains after a given row and column have been taken out of a matrix. An adjoint is simply the transpose of the matrix of cofactors. The real value of the two concepts lies in their usefulness in forming the inverse of a matrix—something that we investigate in the next section.
To compute the matrix of cofactors for a particular matrix, we follow six steps:
Select an element in the original matrix.
Draw a line through the row and column of the element selected. The numbers uncovered represent the cofactor for that element.
Calculate the value of the determinant of the cofactor.
Add together the location numbers of the row and column crossed out in step 2. If the sum is even, the sign of the determinant’s value (from step 3) does not change. If the sum is an odd number, change the sign of the determinant’s value.
The number just computed becomes an entry in the matrix of cofactors; it is located in the same position as the element selected in step 1.
Return to step 1 and continue until all elements in the original matrix have been replaced by their cofactor values.
Let’s compute the matrix of cofactors, and then the adjoint, for the following matrix, using Table M5.1 to help in the calculations:
ELEMENT REMOVED | COFACTORS | DETERMINANT OF COFACTORS | VALUE OF COFACTOR |
---|---|---|---|
Row 1, column 1 | (sign not changed) | ||
Row 1, column 2 | (sign changed) | ||
Row 1, column 3 | 2 (sign not changed) | ||
Row 2, column 1 | (sign changed) | ||
Row 2, column 2 | 4 (sign not changed) | ||
Row 2, column 3 | 25 (sign changed) | ||
Row 3, column 1 | 21 (sign not changed) | ||
Row 3, column 2 | 1 (sign changed) | ||
Row 3, column 3 | (sign not changed) |