Problem 31 - Algebra for College Students

Question

For Problems 21-36, use the technique discussed in this section to find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rrr} 2 & 3 & -4 \\ 3 & -1 & -2 \\ 1 & -4 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix does not have an inverse because its determinant is zero.

Step 1: Understand the Concept

The multiplicative inverse of a matrix, also known as the inverse matrix, is a matrix that, when multiplied by the original matrix, results in the identity matrix. Only square matrices have inverses, and a matrix has an inverse if and only if its determinant is non-zero.

Step 2: Calculate the Determinant

Find the determinant of matrix \( A = \left[ \begin{array}{rrr} 2 & 3 & -4 \ 3 & -1 & -2 \ 1 & -4 & 2 \end{array} \right] \). The determinant for a 3x3 matrix \( A \) is given by \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]Substitute the values:\[ \det(A) = 2((-1)(2) - (-2)(-4)) - 3((3)(2) - (-2)(1)) - 4((3)(-4) - (-1)(1)) \]\[ = 2(-2 - 8) - 3(6 + 2) - 4(-12 + 1) \] \[ = 2(-10) - 3(8) - 4(-11) \] \[ = -20 - 24 + 44 \] \[ = 0 \].

Step 3: Interpret the Determinant Result

Since the determinant is 0, the matrix does not have a multiplicative inverse. A non-zero determinant is required for a matrix to have an inverse.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a unique number that provides useful information about the matrix, specifically regarding its invertibility. It is a scalar value that can be calculated from the elements of a square matrix, and it plays a crucial role in determining if a matrix has an inverse.

For a 3x3 matrix such as: \[A = \left[ \begin{array}{rrr} 2 & 3 & -4 \ 3 & -1 & -2 \ 1 & -4 & 2 \end{array} \right]\]

The determinant \(\det(A)\) is calculated using the formula:

• \(\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\)

Substituting the actual numbers from matrix \(A\) into this formula gives:

• \(\det(A) = 2((-1)(2) - (-2)(-4)) - 3((3)(2) - (-2)(1)) - 4((3)(-4) - (-1)(1))\)
• = 2(-2 - 8) - 3(6 + 2) - 4(-12 + 1)
• = -20 - 24 + 44

This results in a determinant of 0. A zero determinant indicates that the matrix does not have an inverse, which is essential when attempting to find the inverse matrix.

Matrix Multiplication

Matrix multiplication is a fundamental operation that involves combining two matrices to produce another matrix. It is important in numerous applications in mathematics and computer science, including finding inverse matrices. However, it must be noted that matrix multiplication is not always commutative, meaning that the order in which you multiply matrices matters.

When multiplying two matrices \(A\) and \(B\), the resulting matrix \(C = AB\) is formed by taking the dot product of the rows from matrix \(A\) with the columns from matrix \(B\). Each element in matrix \(C\) is the sum of products of corresponding elements from these rows and columns.

To check if a matrix \(A\) has an inverse \(A^{-1}\), we need to ensure that:

• \(AA^{-1} = I\)

where \(I\) is the identity matrix. Only when the determinant is non-zero, as discussed previously, can you compute \(A^{-1}\) such that this matrix multiplication results in the identity matrix.

Identity Matrix

The identity matrix is a special kind of matrix that acts like the number 1 in matrix algebra. When any matrix is multiplied by the identity matrix, the original matrix remains unchanged. In the context of finding an inverse, the identity matrix serves as the confirmation point for achieving an inverse.

For a 3x3 identity matrix \(I\), it looks like this:
\[I = \left[\begin{array}{ccc}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1\end{array}\right]\]

The identity matrix is part of the matrix multiplication process when verifying if a matrix is invertible. For a matrix \(A\) to have an inverse \(A^{-1}\), it must satisfy the equation:

• \(AA^{-1} = I\)

If a matrix, when multiplied by another (its supposed inverse), results in the identity matrix, it confirms that the two matrices are indeed inverses of each other. However, if a matrix's determinant is zero, like the one discussed, it cannot have a multiplicative counterpart to produce an identity matrix.