Problem 49 - Precalculus

Question

determine whether \(B\) is the multiplicative inverse of \(A\) using \(A A^{-1}=I\) $$A=\left[\begin{array}{rrr}1 & -1 & 1 \\\1 & 0 & -1 \\\0 & 1 & -1\end{array}\right] \quad B=\left[\begin{array}{rrr}1 & 0 & 1 \\\1 & -1 & 2 \\\1 & -1 & 1\end{array}\right]$$

Step-by-Step Solution

Answer

No, \( B \) is not the inverse of \( A \) because their product is not the identity matrix.

Step 1: Review Basic Concepts

Before determining if matrix \( B \) is the multiplicative inverse of matrix \( A \), recall that for matrix \( B \) to be the inverse of \( A \), the product \( A \times B \) must equal the identity matrix \( I \). For a 3x3 matrix, \( I \) looks like \( \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \).

Step 2: Calculate the Product \( A \times B \)

Compute the product \( A \times B \):\[ A = \begin{bmatrix} 1 & -1 & 1 \ 1 & 0 & -1 \ 0 & 1 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 & 1 \ 1 & -1 & 2 \ 1 & -1 & 1 \end{bmatrix} \] Perform the matrix multiplication:- Compute the first row: \[ \begin{bmatrix} 1 \times 1 + (-1) \times 1 + 1 \times 1, & 1 \times 0 + (-1) \times (-1) + 1 \times (-1), & 1 \times 1 + (-1) \times 2 + 1 \times 1 \end{bmatrix} \] Which simplifies to \( \begin{bmatrix} 1, & 2, & 0 \end{bmatrix} \).- Compute the second row: \[ \begin{bmatrix} 1 \times 1 + 0 \times 1 + (-1) \times 1, & 1 \times 0 + 0 \times (-1) + (-1) \times (-1), & 1 \times 1 + 0 \times 2 + (-1) \times 1 \end{bmatrix} \] Which simplifies to \( \begin{bmatrix} 0, & 1, & 0 \end{bmatrix} \).- Compute the third row: \[ \begin{bmatrix} 0 \times 1 + 1 \times 1 + (-1) \times 1, & 0 \times 0 + 1 \times (-1) + (-1) \times (-1), & 0 \times 1 + 1 \times 2 + (-1) \times 1 \end{bmatrix} \] Which simplifies to \( \begin{bmatrix} 0, & 0, & 1 \end{bmatrix} \).

Step 3: Compare Result to Identity Matrix

The result of \( A \times B \) is: \[ \begin{bmatrix} 1 & 2 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \] Compare this with the identity matrix \( I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \). The matrices are not equal.

Key Concepts

Identity Matrix

An identity matrix is a special type of square matrix, which acts like the number 1 in matrix multiplication. Just like any number multiplied by 1 remains unchanged, any matrix multiplied by an identity matrix gives the original matrix back. An identity matrix has 1's on the main diagonal (from the top left to the bottom right) and 0's elsewhere.

For example, the identity matrix for a 3x3 matrix is written as:\[I = \begin{bmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1\end{bmatrix}\]

When you multiply any 3x3 matrix by this identity matrix, the original matrix remains unchanged. More importantly, if you multiply a matrix by its inverse, the result is this identity matrix. This property is crucial to understanding matrix inversion, as it helps verify whether two matrices are multiplicative inverses of each other.

Matrix Multiplication

Matrix multiplication is a fundamental operation used quite often in linear algebra. Unlike regular arithmetic multiplication, matrix multiplication is not a straightforward element-by-element operation. Here's how you perform it:

• Ensure that the number of columns in the first matrix equals the number of rows in the second matrix. Only then can you multiply them.
• The element in the resulting matrix is found by taking the dot product of the rows from the first matrix with the columns of the second matrix.

For instance, if multiplying a 3x3 matrix \(A\) by another 3x3 matrix \(B\), you take the first row of matrix \(A\) and multiply each element by the corresponding element in the first column of matrix \(B\), summing all these products to find the first element of the resulting matrix.

Correct matrix multiplication and understanding each element's calculation is vital when determining whether one matrix is the inverse of another. In our exercise, if the product of matrices \(A\) and \(B\) were equal to the identity matrix, then \(B\) would indeed be the inverse of \(A\).

Inverse Matrix

Finding the inverse of a matrix involves determining another matrix that, when multiplied with the original matrix, results in the identity matrix. This inverse only exists for square matrices where the determinant is non-zero.

Here's the key to understanding matrix inversion:

• The inverse of a matrix \(A\) is denoted as \(A^{-1}\).
• The condition for a matrix \(B\) to be the inverse of \(A\) is that both \(A \times B\) and \(B \times A\) equal the identity matrix \(I\).
• To find out if \(B\) is the inverse of \(A\), perform the multiplication and compare with the identity matrix.

In the exercise provided, when matrices \(A\) and \(B\) were multiplied, the result was not the identity matrix, indicating that \(B\) is not the inverse of \(A\). Understanding the properties of inverse matrices helps in solving various problems in linear algebra, especially systems of equations.