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Chapter 3: Problem 69

Evaluate the determinants to verify the equation. $$\left|\begin{array}{ll} w & x \\ y & z \end{array}\right|=\left|\begin{array}{ll} w & x+c w \\ y & z+c y \end{array}\right|$$

Short Answer

Expert verified
Yes, the determinants of the two matrices are indeed equal, hence verifying the given equation.

Step by step solution

01

Step One: Calculate the first determinant

The determinant of the first 2x2 matrix can be calculated using the following formula: \( det(A) = w*z - x*y \) where A is the first matrix.
02

Step Two: Calculate the second determinant

The determinant of the second 2x2 matrix can be calculated using the similar formula but the elements have to be substituted from the second matrix: \( det(B) = w*(z + c*y) - (x + c*w)*y \). This simplifies to \( w*z + c*w*y - x*y - c*w*y \), which simplifies further to \( w*z - x*y \), which is the determinant of the first matrix.
03

Step Three: Compare the two determinants

By comparing the determinants, it can be seen that they are equal. Therefore, the original equation is verified. The determinant of the first matrix equals the determinant of the second matrix, so: \( det(A) = det(B) \).

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Most popular questions from this chapter

Use Cramer's Rule to solve the system of linear equations, if possible. $$\begin{array}{l} 2 x_{1}+3 x_{2}+5 x_{3}=4 \\ 3 x_{1}+5 x_{2}+9 x_{3}=7 \\ 5 x_{1}+9 x_{2}+17 x_{3}=13 \end{array}$$

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Let \(A_{11}, A_{12},\) and \(A_{22}\) be \(n \times n\) matrices. Find the determinant of the partitioned matrix $$\left[\begin{array}{cc}A_{11} & A_{12} \\ 0 & A_{22}\end{array}\right]$$ in terms of the determinants of \(A_{11}, A_{12},\) and \(A_{22}\)

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