Problem 12 - Algebra for College Students

Question

For Problems \(9-20\), find \(A B\) and \(B A\), whenever they exist. $$ A=\left[\begin{array}{rrr} 3 & -1 & -4 \\ -5 & 2 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & -2 \\ -4 & -1 \end{array}\right] $$

Step-by-Step Solution

Answer

\(AB\) does not exist; \(BA = \begin{bmatrix} 19 & -7 & -16 \\ -7 & 2 & 14 \end{bmatrix}\).

Step 1: Determine Compatibility for Matrix Multiplication

Check if the matrices can be multiplied. Matrix \(A\) is a \(2\times 3\) matrix and matrix \(B\) is a \(2\times 2\) matrix. For the product \(AB\) to exist, the number of columns in \(A\) (which is 3) must equal the number of rows in \(B\) (which is 2). Therefore, \(AB\) does not exist because they are not compatible in size for multiplication.

Step 2: Check Compatibility for the Reverse Multiplication \(BA\)

Matrix \(B\) is \(2\times 2\) and matrix \(A\) is \(2\times 3\). For \(BA\), the number of columns in \(B\) (which is 2) must equal the number of rows in \(A\) (which is 2).\(BA\) is a valid operation.

Step 3: Perform Matrix Multiplication \(BA\)

Calculate the product \(BA\):\[BA = \begin{bmatrix} 3 & -2 \ -4 & -1 \end{bmatrix} \begin{bmatrix} 3 & -1 & -4 \ -5 & 2 & 2 \end{bmatrix} \]To find the element in the first row, first column:\\(3 \times 3 + (-2) \times (-5) = 9 + 10 = 19\)\To find the element in the first row, second column:\\(3 \times (-1) + (-2) \times 2 = -3 -4 = -7\)\To find the element in the first row, third column:\\(3 \times (-4) + (-2) \times 2 = -12 - 4 = -16\)Repeat for the second row:To find the element in the second row, first column:\\(-4 \times 3 + (-1) \times (-5) = -12 + 5 = -7\)To find the element in the second row, second column:\\(-4 \times (-1) + (-1) \times 2 = 4 - 2 = 2\)To find the element in the second row, third column:\\(-4 \times (-4) + (-1) \times 2 = 16 - 2 = 14\)Thus, \(BA = \begin{bmatrix} 19 & -7 & -16 \ -7 & 2 & 14 \end{bmatrix}\).

Step 4: Conclusion

The product \(AB\) does not exist, but \(BA\) does and is calculated as \(\begin{bmatrix} 19 & -7 & -16 \ -7 & 2 & 14 \end{bmatrix}\).

Key Concepts

Matrix Compatibility

When it comes to multiplying two matrices, the first thing you need to do is check if they are compatible for multiplication. This concept of "Matrix Compatibility" refers to specific conditions that matrices must meet for their product to exist.
Matrix dimensions play a crucial role here. You must look at the number of columns in the first matrix and the number of rows in the second matrix.
In our example:

• Matrix \(A\) is a \(2 \times 3\) matrix, meaning it has 2 rows and 3 columns.
• Matrix \(B\) is a \(2 \times 2\) matrix, which has 2 rows and 2 columns.

For the product \(AB\) to exist, the number of columns in \(A\) (3) must equal the number of rows in \(B\) (2). Here, they are not equal, so \(AB\) does not exist.
However, for \(BA\) to exist, the number of columns in \(B\) (2) must match the number of rows in \(A\) (2). Since this condition is met, \(BA\) can be calculated.

Product of Matrices

After confirming the compatibility of matrices for multiplication, the next step is finding their product. "Product of Matrices" involves multiplying each element of the rows from the first matrix by the corresponding elements of the columns from the second matrix.Let’s compute \(BA\):
To find the first element of the resulting matrix:

• Multiply the first row of \(B\) by the first column of \(A\).
• This is done by calculating \((3 \times 3) + ((-2) \times (-5)) = 9 + 10 = 19\).

To find the second element in the first row of the resulting matrix:

• Multiply the first row of \(B\) by the second column of \(A\): \((3 \times (-1)) + ((-2) \times 2) = -3 -4 = -7\).

Continue the same process for each element. The computation for \(BA\) is completed when we perform these operations for rows and columns appearing in both matrices.

Matrix Dimensions

Understanding matrix dimensions is essential when discussing matrix multiplication. The "Matrix Dimensions" indicate the size and shape of the matrix, which are determined by the number of rows and columns it contains.For instance:
Matrix \(A\) has dimensions \(2 \times 3\), meaning:

• It has 2 rows.
• It has 3 columns.

Matrix \(B\), with dimensions \(2 \times 2\), includes:

• 2 rows.
• 2 columns.

When attempting to multiply matrices, the column count of the first matrix must match the row count of the second, which wasn't satisfied in the case of \(AB\) but was for \(BA\).
The resulting matrix from \(BA\), having dimensions derived from the outer dimensions of matrices \(B\) and \(A\), is \(2 \times 3\), indicating 2 rows and 3 columns in the final product.