Problem 31 - Precalculus Mathematics for Calculus

Question

Let $$B=\left[\begin{array}{rrr} 4 & 1 & 0 \\ -2 & -1 & 1 \\ 4 & 0 & 3 \end{array}\right]$$ (a) Evaluate \(\operatorname{det}(B)\) by expanding by the second row. (b) Evaluate \(\operatorname{det}(B)\) by expanding by the third column. (c) Do your results in parts (a) and (b) agree?

Step-by-Step Solution

Answer

No, the results do not agree. The first method gives -22, the second gives 2.

Step 1: Setup for Row Expansion

To evaluate \(\operatorname{det}(B)\) by expanding along the second row, identify the elements of the second row: \(-2, -1, 1\). Write the formula for determinant expansion along a row: \(\operatorname{det}(B) = a_{2,1}C_{2,1} + a_{2,2}C_{2,2} + a_{2,3}C_{2,3}\), where \(a_{i,j}\) is the element of the matrix and \(C_{i,j}\) is the cofactor of that element.

Step 2: Calculate Cofactors for Row Expansion

Compute the cofactors for each of the elements in the second row. - For \(a_{2,1} = -2\), the cofactor \(C_{2,1} = (-1)^{3} \cdot \begin{vmatrix} 1 & 0 \ 0 & 3 \end{vmatrix} = 3 \).- For \(a_{2,2} = -1\), the cofactor \(C_{2,2} = (-1)^{4} \cdot \begin{vmatrix} 4 & 0 \ 4 & 3 \end{vmatrix} = 12 \).- For \(a_{2,3} = 1\), the cofactor \(C_{2,3} = (-1)^{5} \cdot \begin{vmatrix} 4 & 1 \ 4 & 0 \end{vmatrix} = -4 \).

Step 3: Compute Determinant by Row Expansion

Substitute the computed cofactors into the expansion formula: \(\operatorname{det}(B) = -2 \cdot 3 + (-1) \cdot 12 + 1 \cdot (-4) = -6 - 12 - 4 = -22\).

Step 4: Setup for Column Expansion

To evaluate \(\operatorname{det}(B)\) by expanding along the third column, identify the elements of the third column: \(0, 1, 3\). Use the formula for determinant expansion along a column: \(\operatorname{det}(B) = a_{1,3}C_{1,3} + a_{2,3}C_{2,3} + a_{3,3}C_{3,3}\).

Step 5: Calculate Cofactors for Column Expansion

Compute the cofactors for each of the elements in the third column.- For \(a_{1,3} = 0\), any cofactor multiplied by \(0\) will remain \(0\).- For \(a_{2,3} = 1\), the cofactor \(C_{2,3} = (-1)^{5} \cdot \begin{vmatrix} 4 & 1 \ 4 & 0 \end{vmatrix} = -4\).- For \(a_{3,3} = 3\), the cofactor \(C_{3,3} = (-1)^{6} \cdot \begin{vmatrix} 4 & 1 \ -2 & -1 \end{vmatrix} = 2\).

Step 6: Compute Determinant by Column Expansion

Substitute the computed cofactors into the expansion formula: \(\operatorname{det}(B) = 0 \cdot C_{1,3} + 1 \cdot (-4) + 3 \cdot 2 = -4 + 6 = 2\).

Step 7: Compare Results of Both Methods

Compare the determinant values obtained from both methods. From row expansion, we found \(-22\), while from column expansion, we found \(2\). Since these values do not match, verify calculations for errors.

Key Concepts

Row Expansion

Row expansion is a method of calculating the determinant of a matrix by focusing on a specific row. When you perform row expansion, you select one row of the matrix and multiply each element in that row by its corresponding cofactor, and then add or subtract these products. For a matrix element denoted by \(a_{i,j}\), the expansion can be expressed as:

• \(\text{det}(B) = a_{i,1}C_{i,1} + a_{i,2}C_{i,2} + a_{i,3}C_{i,3}\), where each \(C_{i,j}\) is a cofactor.

To compute the determinant of the matrix \(B\) given in the original exercise by the second row, identify its elements: \(-2, -1,\) and \(1\). Each element needs to be multiplied by its respective cofactor, which is then alternately added or subtracted according to their position (considering the sign pattern from the checkerboard method of determinant calculation).
For the second row in the original solution:

• The first element is \(-2\) with a cofactor of \(3\), leading to \(-2 \cdot 3\).
• The second element is \(-1\), with its cofactor being \(12\).
• The third element is \(1\), and its cofactor is \(-4\).

So, the row expansion method yields a determinant of \(-22\), using these calculations.

Column Expansion

Column expansion shares similar principles with row expansion. However, instead of focusing on a row, you focus on a specific column of the matrix. Again, each element of the chosen column is multiplied by its cofactor.

• The formula for column expansion is: \(\text{det}(B) = a_{1,j}C_{1,j} + a_{2,j}C_{2,j} + a_{3,j}C_{3,j}\).

In the given problem, the third column's elements are \(0, 1,\) and \(3\). Each element is multiplied by its cofactor:

• If an element is zero, the entire term becomes zero regardless of the cofactor.
• For \(1\), the corresponding cofactor is \(-4\).
• The element \(3\) multiplies its cofactor, which is \(2\).

The result of this column expansion comes out as \(2\). Although systematic errors can sometimes lead to different outcomes, you should usually get the same determinant result from both row and column expansions.

Cofactor

Cofactors play a crucial role in calculating determinants through row or column expansions. Each element of the matrix has an associated cofactor, determined by a minor matrix. A minor is what you get from removing the element's row and column, allowing you to find the determinant of the remaining submatrix.

• The cofactor \(C_{i,j}\) is calculated as \((-1)^{i+j}\) times the determinant of the corresponding minor matrix.

Here's how you compute cofactors:

• Ever wondered why there's a sign pattern employed in cofactor calculations? It derives from the \((-1)^{i+j}\) factor. If the sum of the indices \((i+j)\) is even, the sign remains positive; if odd, it becomes negative.
• The size of the minor matrix depends on the original matrix and is one size smaller since you remove one row and one column for each element's cofactor.

Understanding cofactors is critical because they directly affect the sign and value of your determinant result in both row and column expansions. It's important to calculate them correctly to ensure accurate determinant computation.