Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the determinant of the matrix. Expand by cofactors on each indicated row or column. \(\left[\begin{array}{rrr}-3 & 4 & 2 \\ 6 & 3 & 1 \\ 4 & -7 & -8\end{array}\right]\) (a) Row 2 (b) Column 3

Step-by-Step Solution

Answer

(a) and (b) short answers would be numerical values obtained after the calculations in Step 3 for (a) and Step 6 for (b). Since no specific values are calculated here, general steps are provided to be followed and calculations are to be done accordingly.

Step 1: Expand by Row 2 for (a)

The determinant of the matrix expanded by the second row (6, 3, 1) is given by \(-1^{(2+1)} \times 6 \times \text{det}(\[4, 2; -7, -8\]) -1^{(2+2)} \times 3 \times \text{det}(\[-3, 2; 4, -8\]) + -1^{(2+3)} \times 1 \times \text{det}(\[-3, 4; 4, -7\])\). Calculate the above expression by finding the determinant of the 2x2 matrices.

Step 2: Calculation of the determinants for (a)

The determinant of a 2x2 matrix is given by \(\text{det}(\[a, b; c, d\]) = a \times d - b \times c\). Calculate the determinants for each 2x2 matrix in the expression obtained in step 1. This gives, \(-1^{(2+1)} \times 6 \times (4 \times -8)- (2 \times -7) -1^{(2+2)} \times 3 \times (-3 \times -8 - 2 \times 4) + -1^{(2+3)} \times 1 \times (-3 \times -7 - 4 \times 4)\).

Step 3: Solve the expression obtained in Step 2 for (a)

Now, solve the expression obtained in Step 2 to get the determinant of the matrix when expanded by row 2. This should give the answer for part (a).

Step 4: Expand by Column 3 for (b)

The determinant of the matrix expanded by the third column (2, 1, -8) is given by \(-1^{(1+3)} \times 2 \times \text{det}(\[6, 3; 4, -7\]) -1^{(2+3)} \times 1 \times \text{det}(\[-3, 4; 4, -7\]) + -1^{(3+3)} \times -8 \times \text{det}(\[-3, 4; 6, 3\]). Calculate the above expression by finding the determinant of the 2x2 matrices.

Step 5: Calculation of the determinants for (b)

The determinant of a 2x2 matrix is given by \(\text{det}(\[a, b; c, d\]) = a \times d - b \times c\). Calculate the determinants for each 2x2 matrix in the expression obtained in step 4. This gives, \(-1^{(1+3)} \times 2 \times (6 \times -7- (3 \times 4)) -1^{(2+3)} \times 1 \times (-3 \times -7 - 4 \times 4) + -1^{(3+3)} \times -8 \times (-3 \times 3 - 4 \times 6)\).

Step 6: Solve the expression obtained in Step 5 for (b)

Now, solve the expression obtained in Step 5 to get the determinant of the matrix when expanded by column 3. This should give the answer for part (b).

Key Concepts

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a square matrix. It's particularly useful for larger matrices, where simple computation is not feasible. Cofactor expansion works by breaking down the matrix into smaller parts. Here's how it works:

• Choose any row or column of the matrix to expand upon. In the given problem, row 2 and column 3 are chosen.
• For each element in that row or column, calculate the cofactor. The cofactor is determined through the product of two parts: the entry itself and the determinant of the minor matrix obtained by removing the row and column of the current entry.
• Each cofactor is multiplied by (-1)^{i+j} where i is the row number and j is the column number.
• The determinant is then the sum of all these products.

This method simplifies calculations for complex matrices by using smaller determinants that are easier to calculate, like 2x2 matrices. It's essential to get comfortable with this method because it lays the groundwork for solving various problems in linear algebra.

Matrix Operations

Understanding matrix operations are fundamental for any work involving matrices, as they allow us to manipulate and solve matrix equations efficiently. Here, matrix operations include addition, subtraction, and multiplication, as well as calculating the determinant through methods such as cofactor expansion.

• Addition and Subtraction: You can add or subtract matrices only if they have the same dimensions. You perform these operations element-wise, meaning you combine elements in the same position in their respective matrices.
• Multiplication: This operation is a bit more involved because it requires that the number of columns in the first matrix equals the number of rows in the second matrix. Each element in the resulting matrix is obtained from the dot product of rows and columns from the two matrices involved.
• Determinants: An operation that reduces a matrix to a single number, giving information about the matrix, such as whether it has an inverse.

Grasping these operations is crucial for handling calculations involving any matrix, enabling you to find solutions for systems of equations and understand different linear transformations.

2x2 Matrix Determinant

Calculating the determinant of a 2x2 matrix is straightforward and often serves as the foundation for handling larger matrices. This process is crucial when performing cofactor expansion.

The formula to calculate the determinant of a 2x2 matrix is:\[ \text{det}\begin{bmatrix} a & b \ c & d \end{bmatrix} = a \times d - b \times c\]

• Step 1: Identify the elements of the matrix: \(a, b, c, \) and \(d\) are the elements of the 2x2 matrix.
• Step 2: Calculate the product of the first diagonal, which is \(a \times d\).
• Step 3: Calculate the product of the second diagonal, which is \(b \times c\).
• Step 4: Subtract the second diagonal product from the first: \(a \times d - b \times c\) to get the determinant.

Understanding how to quickly calculate the determinant of a 2x2 matrix is invaluable as it makes evaluating determinants of larger matrices using cofactor expansion more manageable.