Problem 3 - College Algebra

Question

find the products \(A B\) and \(B A\) to determine whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{rr} -4 & 0 \\ 1 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} -2 & 4 \\ 0 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

No, matrix \(B\) is not the multiplicative inverse of matrix \(A\) as neither \(AB\) nor \(BA\) is equal to the identity matrix \(I\).

Step 1: Matrix multiplication A B

Multiply matrix \(A\) by matrix \(B\).\n\n\(AB=\left[\begin{array}{rr} -4 & 0 \ 1 & 3 \end{array}\right] * \left[\begin{array}{rr} -2 & 4 \ 0 & 1 \end{array}\right] \)\n\nCalculate the element at row 1 column 1 for \(AB\) by performing scalar multiplication and adding up the results with corresponding elements in row 1 of matrix \(A\) and column 1 of matrix \(B\). Repeat the process for remaining elements.\n\n\(AB = \left[\begin{array}{rr} 8 & -16 \ -2 & 7 \end{array}\right]\)

Step 2: Matrix multiplication B A

Multiply matrix \(B\) by matrix \(A\).\n\n\(BA = \left[\begin{array}{rr} -2 & 4 \ 0 & 1 \end{array}\right] * \left[\begin{array}{rr} -4 & 0 \ 1 & 3 \end{array}\right]\)\n\nCalculate the element at row 1 column 1 for \(BA\) by performing scalar multiplication and adding up the results with corresponding elements in row 1 of matrix \(B\) and column 1 of matrix \(A\). Repeat the process for remaining elements.\n\n\(BA = \left[\begin{array}{rr} 8 & 12 \ 1 & 3 \end{array}\right]\)

Step 3: Compare with Identity Matrix

To determine whether \(B\) is the multiplicative inverse of \(A\), both results (\(AB\) and \(BA\)) should be the identity matrix \(I\).\n\nHere, \(I = \left[\begin{array}{rr} 1 & 0 \ 0 & 1 \end{array}\right]\)\n\nHowever, neither \(AB\) nor \(BA\) is equal to \(I\). Therefore, we can conclude that \(B\) is not the multiplicative inverse of \(A\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that combines two matrices to produce a new matrix. The process involves taking the rows of the first matrix (known as the left matrix) and the columns of the second matrix (the right matrix) and performing dot products to determine the elements of the result matrix.

For the dot product in each position, you multiply the elements of the row by the corresponding elements of the column and then add all those products together. This is repeated for each element in the resulting matrix.

Requirements for Multiplication

• The number of columns in the first matrix must equal the number of rows in the second matrix.
• The resulting matrix will have dimensions equal to the number of rows in the first matrix and the number of columns in the second matrix.

For instance, if matrix A is of size 2x2 and matrix B is also of size 2x2, you can multiply them to produce another 2x2 matrix, provided their multiplication aligns by rows and columns. If the matrices don't meet the requirement for multiplication (i.e., mismatched rows and columns), the multiplication process is undefined and cannot be performed.

Identity Matrix

The identity matrix, usually denoted as I, is a square matrix with ones on the main diagonal and zeroes everywhere else. It acts as the multiplicative identity in the set of square matrices, much like the number 1 in regular multiplication.

Here's why the identity matrix is significant:

• When you multiply any matrix A by the identity matrix (on the left or the right), you get the original matrix A back. It's like multiplying a number by 1.
• The identity matrix serves as a benchmark for finding the multiplicative inverse of a matrix, which is a matrix that when multiplied with the original matrix, yields the identity matrix.

For a 2x2 matrix, the identity matrix is I = [[1, 0], [0, 1]]. In the context of our problem, we compare the result of the matrix multiplication to this identity matrix to verify if one matrix is indeed the inverse of the other.

Inverse Matrix

An inverse matrix for a given square matrix A is another matrix, often denoted as A^-1, that, when multiplied with A, produces the identity matrix. It's similar to division in regular arithmetic, but since division isn't defined for matrices, we use the concept of inversion.

Here's how it works:

• A * A^-1 = I and A^-1 * A = I, where I is the identity matrix.
• If A^-1 exists, A is said to be invertible or non-singular, meaning it has an inverse. Not all matrices have inverses.
• The multiplicativity of matrix inverses must hold true, meaning AB = BA = I for B to be considered the inverse of A.

In our exercise, the matrix B was tested to check if it's the inverse of matrix A. After performing the multiplication, we found that neither AB nor BA resulted in the identity matrix, indicating that B is not the inverse of A.