Problem 40 - College Algebra with Modeling and Visualization

Question

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rr}3 & -1 \\\2 & -2 \\\0 & 4\end{array}\right], \quad B=\left[\begin{array}{rrr}1 & -4 & 0 \\\\-1 & 3 & 2\end{array}\right]$$

Step-by-Step Solution

Answer

The product \(A B = \left[\begin{array}{ccc}4 & -15 & -2 \\4 & -14 & -4 \\-4 & 12 & 8\end{array}\right]\). The product \(B A = \left[\begin{array}{cc}-5 & 7 \\3 & 3\end{array}\right]\).

Step 1: Check Matrix Dimensions for Multiplication

The matrix multiplication is possible between two matrices if the number of columns in the first matrix is equal to the number of rows in the second matrix. Matrix \(A\) is of size \(3 \times 2\) and Matrix \(B\) is of size \(2 \times 3\). Thus, \(A B\) is possible because the number of columns in \(A\) (2) matches the number of rows in \(B\) (2).

Step 2: Matrix Multiplication A B

To find \(A B\), we perform matrix multiplication. Each element \(c_{ij}\) in the resulting matrix \(C\) is calculated as: \[ c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj} \]For \(A B\), - First Row: - \(c_{11} = (3)(1) + (-1)(-1) = 3 + 1 = 4\) - \(c_{12} = (3)(-4) + (-1)(3) = -12 - 3 = -15\) - \(c_{13} = (3)(0) + (-1)(2) = 0 - 2 = -2\)- Second Row: - \(c_{21} = (2)(1) + (-2)(-1) = 2 + 2 = 4\) - \(c_{22} = (2)(-4) + (-2)(3) = -8 - 6 = -14\) - \(c_{23} = (2)(0) + (-2)(2) = 0 - 4 = -4\)- Third Row: - \(c_{31} = (0)(1) + (4)(-1) = 0 - 4 = -4\) - \(c_{32} = (0)(-4) + (4)(3) = 0 + 12 = 12\) - \(c_{33} = (0)(0) + (4)(2) = 0 + 8 = 8\)Therefore, \(A B = \left[\begin{array}{ccc}4 & -15 & -2 \4 & -14 & -4 \-4 & 12 & 8\end{array}\right]\) .

Step 3: Check Dimensions for B A

Now, consider the multiplication \(B A\). Matrix \(B\) is \(2 \times 3\) and matrix \(A\) is \(3 \times 2\), so \(B A\) is possible as well since the number of columns in \(B\) (3) matches the number of rows in \(A\) (3).

Step 4: Matrix Multiplication B A

Perform the multiplication for \(B A\):- First Row: - \(c_{11} = (1)(3) + (-4)(2) + (0)(0) = 3 - 8 + 0 = -5\) - \(c_{12} = (1)(-1) + (-4)(-2) + (0)(4) = -1 + 8 + 0 = 7\)- Second Row: - \(c_{21} = (-1)(3) + (3)(2) + (2)(0) = -3 + 6 + 0 = 3\) - \(c_{22} = (-1)(-1) + (3)(-2) + (2)(4) = 1 - 6 + 8 = 3\)Thus, \(B A = \left[\begin{array}{cc}-5 & 7 \3 & 3\end{array}\right]\) .

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial when dealing with matrix multiplication. Each matrix has a particular number of rows and columns, defined as its dimensions. For example, matrix \( A \) has dimensions \( 3 \times 2 \), meaning it has 3 rows and 2 columns. Similarly, matrix \( B \) is \( 2 \times 3 \), with 2 rows and 3 columns.

For matrix multiplication to be possible, the number of columns in the first matrix must match the number of rows in the second matrix. This condition ensures that each element in the resulting matrix is calculated properly. In our example, since matrix \( A \) has 2 columns and matrix \( B \) has 2 rows, you can multiply \( A \) by \( B \). Conversely, because \( B \) has 3 columns and \( A \) has 3 rows, you can also multiply \( B \) by \( A \).

Always remember to verify matrix dimensions before attempting multiplication:

• Matrix \( A \) dimensions: \( 3 \times 2 \)
• Matrix \( B \) dimensions: \( 2 \times 3 \)
• Multiplication \( A \cdot B \) is possible
• Multiplication \( B \cdot A \) is also possible

Resulting Matrix

After verifying the dimensions, the next step in matrix multiplication is identifying the resulting matrix's size. The dimensions of the resultant matrix are derived from the outer dimensions of the two original matrices. For the multiplication \( A \cdot B \) where matrix \( A \) is \( 3 \times 2 \) and matrix \( B \) is \( 2 \times 3 \), the resulting matrix will have the dimensions \( 3 \times 3 \) because:

• The number of rows comes from the first matrix \( A \) (3 rows).
• The number of columns comes from the second matrix \( B \) (3 columns).

When running the calculations for \( B \cdot A \), the resulting matrix will be \( 2 \times 2 \). Here, the 2 rows are taken from matrix \( B \) and the 2 columns from matrix \( A \).

Thus, understanding the dimensions of the resulting matrix helps in visualizing how the final matrix will look and ensures that calculations are consistently applied throughout the solving process.

Multiplication Steps

Now, let's delve into the essential part: the actual multiplication steps. It's important to understand how each element in the resulting matrix is computed using the elements of the original matrices. Calculate each element by taking the dot product of the corresponding row of the first matrix and the column of the second matrix.

In the example problem, for \( A \cdot B \):

• First Row:
  • Element \( c_{11} \) is \( (3 \times 1) + (-1 \times -1) = 4 \)
  • Element \( c_{12} \) is \( (3 \times -4) + (-1 \times 3) = -15 \)
  • Element \( c_{13} \) is \( (3 \times 0) + (-1 \times 2) = -2 \)
• Second Row:
  • Element \( c_{21} \) is \( (2 \times 1) + (-2 \times -1) = 4 \)
  • Element \( c_{22} \) is \( (2 \times -4) + (-2 \times 3) = -14 \)
  • Element \( c_{23} \) is \( (2 \times 0) + (-2 \times 2) = -4 \)
• Third Row:
  • Element \( c_{31} \) is \( (0 \times 1) + (4 \times -1) = -4 \)
  • Element \( c_{32} \) is \( (0 \times -4) + (4 \times 3) = 12 \)
  • Element \( c_{33} \) is \( (0 \times 0) + (4 \times 2) = 8 \)

With these calculations, you've constructed matrix \( C \) for the product \( A \cdot B \), and a similar process will yield the matrix for \( B \cdot A \). Understanding these steps helps ensure success in any matrix multiplication problem you encounter.