Problem 11 - Algebra for College Students

Question

For Problems \(9-20\), find \(A B\) and \(B A\), whenever they exist. $$ A=\left[\begin{array}{llr} 2 & -1 & -3 \\ 0 & -4 & 7 \end{array}\right], \quad B=\left[\begin{array}{rrrr} 2 & 1 & -1 & 4 \\ 0 & -2 & 3 & 5 \\ -6 & 4 & -2 & 0 \end{array}\right] $$

Step-by-Step Solution

Answer

AB exists and is a 2x4 matrix, while BA does not exist.

Step 1: Understand Matrix Multiplication

To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. For matrix \(A\) (which is a \(2 \times 3\) matrix) and matrix \(B\) (which is \(3 \times 4\)), the product \(AB\) can be computed because \(A\) has 3 columns and \(B\) has 3 rows.

Step 2: Calculate AB

To compute \(AB\), multiply each element of the rows of \(A\) by the corresponding element of the columns of \(B\), and sum the products for each element in the resultant matrix. The element at position \((i, j)\) is computed by summing the products of the elements of the \(i\)-th row of \(A\) and the \(j\)-th column of \(B\).

Step 2.1: Element (1, 1) of AB

Compute: \[ 2 \times 2 + (-1) \times 0 + (-3) \times (-6) = 4 + 0 + 18 = 22 \]

Step 2.2: Element (1, 2) of AB

Compute: \[ 2 \times 1 + (-1) \times (-2) + (-3) \times 4 = 2 + 2 - 12 = -8 \]

Step 2.3: Element (1, 3) of AB

Compute: \[ 2 \times -1 + (-1) \times 3 + (-3) \times (-2) = -2 - 3 + 6 = 1 \]

Step 2.4: Element (1, 4) of AB

Compute: \[ 2 \times 4 + (-1) \times 5 + (-3) \times 0 = 8 - 5 + 0 = 3 \]

Step 2.5: Element (2, 1) of AB

Compute: \[ 0 \times 2 + (-4) \times 0 + 7 \times (-6) = 0 + 0 - 42 = -42 \]

Step 2.6: Element (2, 2) of AB

Compute: \[ 0 \times 1 + (-4) \times (-2) + 7 \times 4 = 0 + 8 + 28 = 36 \]

Step 2.7: Element (2, 3) of AB

Compute: \[ 0 \times -1 + (-4) \times 3 + 7 \times (-2) = 0 - 12 - 14 = -26 \]

Step 2.8: Element (2, 4) of AB

Compute: \[ 0 \times 4 + (-4) \times 5 + 7 \times 0 = 0 - 20 + 0 = -20 \]

Step 3: Present Matrix AB

The resulting product for \(AB\) is a \(2 \times 4\) matrix: \[\begin{bmatrix}22 & -8 & 1 & 3 \-42 & 36 & -26 & -20\end{bmatrix}\]

Step 4: Check Existence of BA

To multiply \(BA\), \(B\) should have the same number of columns as the number of rows in \(A\). Since \(B\) is a \(3 \times 4\) matrix and \(A\) is a \(2 \times 3\) matrix, \(BA\) is not defined because the number of columns in \(B\) (4) does not match the number of rows in \(A\) (2).

Key Concepts

Matrix Dimensions

Understanding the dimensions of a matrix is essential when working with matrices, especially for operations like multiplication. The dimensions of a matrix refer to the number of rows and columns it contains. For example, matrix \(A\) in our exercise is a \(2 \times 3\) matrix, which means it has 2 rows and 3 columns. Similarly, matrix \(B\) is a \(3 \times 4\) matrix, indicating 3 rows and 4 columns.

This information about dimensions is crucial because it directly affects whether or not matrices can be multiplied together. The matrix multiplication rule dictates that for two matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In our case, for the matrices \(A\) and \(B\), this condition is satisfied, allowing us to compute the product \(AB\).

It's always a good practice to check the dimensions of matrices before attempting multiplication, as it avoids potential errors and unnecessary calculations.

Matrix Product

The matrix product is a core operation involving matrices and is carried out only if the multiplication condition is met. For matrices \(A\) and \(B\), where \(A\) is \(2 \times 3\) and \(B\) is \(3 \times 4\), we can find the matrix product \(AB\).

To calculate the product \(AB\), follow these steps:

• Multiply the elements of each row in \(A\) by the corresponding elements of each column in \(B\).
• Sum these products to get the elements of the resulting matrix.

For instance, the element in the first row and first column of \(AB\) is computed as the sum of the products of the corresponding row from \(A\) and column from \(B\):
\[ 2 \times 2 + (-1) \times 0 + (-3) \times (-6) = 22 \]
Proceed to calculate each element similarly until the new matrix \[ \begin{bmatrix}22 & -8 & 1 & 3 \ -42 & 36 & -26 & -20 \end{bmatrix} \] is formed, which has dimensions \(2 \times 4\).

The process of matrix multiplication can seem complex, but with practice, it becomes more intuitive. Remember, the resulting matrix always has dimensions that are determined by the "outer" numbers of the original matrices, in this case \(2\) and \(4\), describing the product \(AB\).

Existence of Matrix Product

Not all matrix multiplications result in a valid product. The existence of a matrix product depends on the compatibility of the matrices involved. For two matrices, say \(X\) of size \(m \times n\) and \(Y\) of size \(p \times q\), multiplication is only defined if \(n = p\). This condition means that the number of columns in \(X\) must equal the number of rows in \(Y\) for the multiplication to be possible.

In our exercise, while the product \(AB\) exists, let's consider \(BA\). Matrix \(B\) is \(3 \times 4\) and matrix \(A\) is \(2 \times 3\). To evaluate \(BA\), the number of columns in \(B\) (which is 4) would need to match the number of rows in \(A\) (which is 2). Since these numbers do not match, the product \(BA\) does not exist.

It's helpful to remember that even if \(AB\) is a valid product, \(BA\) is not guaranteed to exist, highlighting the importance of analyzing matrix dimensions before proceeding with matrix multiplication. This property also showcases how matrix multiplication is not commutative, meaning \(AB\) does not necessarily equal \(BA\), and in some cases, \(BA\) may not even be defined.