Practice Makes Perfect Algebra - Carolyn Wheater [38]
Since the matrix is square and the determinant is non-0, we can continue to find an inverse. We create a matrix of the same size, in this case 3 × 3, and fill it with an alternating pattern of pluses and minuses. These are known as position signs.
We return to the original matrix and find the cofactor of each element. The cofactor of an element is found by eliminating the row and column that contain the element and calculating the determinant of the remaining matrix. In our original matrix, we find the cofactor of the 1 in row 1, column 1 by crossing out the first row and first column: . Then we find the determinant . The cofactor of the element 1 is 2. We place this cofactor in the first row, first column of the matrix we filled with position signs: .
We repeat this process for each element in the original matrix, placing the cofactor in the corresponding place in the new matrix of cofactors. The cofactor of the 3 in row 1, column 2 is found by eliminating row 1 and column 2: . Then the determinant is calculated and placed with its position sign. Since the sign for this position is a minus, we have .
By the same method, we find that the cofactor of −2 in row 1, column 3 is , and we place that value in the matrix: .
Repeating these steps for the elements in the second row gives three more cofactors.
These are placed in the corresponding positions in the matrix of cofactors, each with its position sign: .
The same steps are repeated for the elements in the third row.
This gives us the matrix of cofactors . The next step in the process of creating the inverse is to transpose the matrix of cofactors we have just formed. Transposing, or forming the transpose, is accomplished by placing the elements that had been the first row in the first column, the second row in the second column, and so on. In this case the matrix of cofactors is transposed to become . The transpose of the matrix of cofactors is called the adjoint.
The final step in the process of creating the inverse is to multiply the adjoint by the reciprocal of the determinant. The determinant, calculated earlier, was 45, so we multiply by .
The inverse matrix often involves fractions, which cannot always be simplified. Verifying that this is, in fact, the inverse can involve some cumbersome arithmetic, and if you are going to perform the multiplication by hand, it may be wise to use the unsimplified version, since you will need a common denominator for the addition. Remember that you must check the product in both directions.
The product in the other direction is similar.
The product in the other direction is similar.
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EXERCISE 13.6
Determine whether the given matrices are inverses.
Tell whether each matrix has an inverse. Do not find the inverse matrix.
Find the inverse of each matrix, if possible.
Solving systems with matrices
Solving a system of equations—two or more equations in two or more variables—is a common task in algebra. In order to arrive at a solution, you need to have as many equations as variables. There are three methods of solving a system of equations that make use of matrix algebra: Cramer’s rule, the method of inverses, and reduced row echelon form.
Cramer’s rule
Cramer’s rule is a method for determining the solution of a system of equations by means of determinants. You probably remember from algebra the elimination method of solving a system of equations. One or both equations can be multiplied by a constant and then the equations added to eliminate one of the variables. Solving the system will allow us to look at the options available. If we choose to eliminate x from the equations, the first equation can be multiplied by 3 and the second equation multiplied by −2.
Adding the equations eliminates x.
Once the value of one variable is known, it can be substituted back into one of the original equations: 2x + 3 · 3 = 19 ⇒ 2x = ⇒ x = 5.
If instead we choose to eliminate y, the first equation is multiplied by 4 and the