Problem 72 - A Graphical Approach to Precalculus with Limits

Question

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right]\), find each product if possible. $$C B$$

Step-by-Step Solution

Answer

The product \(CB\) is \(\left[\begin{array}{cc}-22 & -6 \\ 18 & 36\end{array}\right]\).

Step 1: Verify Matrix Compatibility for Multiplication

To multiply two matrices, the number of columns in the first matrix should match the number of rows in the second matrix. Here, matrix \(C\) is of size \(2 \times 3\), and matrix \(B\) is of size \(3 \times 2\). Since the number of columns in \(C\) (3 columns) matches the number of rows in \(B\) (3 rows), the multiplication is possible.

Step 2: Set Up the Matrix Multiplication

The resulting matrix from multiplying a \(2 \times 3\) matrix by a \(3 \times 2\) matrix will be a \(2 \times 2\) matrix. Let's denote the resulting matrix as \(CB\), where the element in the \(i^{th}\) row and \(j^{th}\) column is given by:\[(CB)_{ij} = \sum_{k=1}^{3} C_{ik} B_{kj}\]

Step 3: Calculate the Element (1,1) of CB

Using the formula, calculate the element in the first row and first column of \(CB\):\[(CB)_{11} = (-5)(5) + (4)(0) + (1)(3)\]\[(CB)_{11} = -25 + 0 + 3 = -22\]

Step 4: Calculate the Element (1,2) of CB

Next, calculate the element in the first row and second column of \(CB\):\[(CB)_{12} = (-5)(1) + (4)(-2) + (1)(7)\]\[(CB)_{12} = -5 - 8 + 7 = -6\]

Step 5: Calculate the Element (2,1) of CB

Calculate the element in the second row and first column of \(CB\):\[(CB)_{21} = (0)(5) + (3)(0) + (6)(3)\]\[(CB)_{21} = 0 + 0 + 18 = 18\]

Step 6: Calculate the Element (2,2) of CB

Finally, calculate the element in the second row and second column of \(CB\):\[(CB)_{22} = (0)(1) + (3)(-2) + (6)(7)\]\[(CB)_{22} = 0 - 6 + 42 = 36\]

Step 7: Write the Resultant Matrix CB

After calculating all elements, the product \(CB\) results in:\[CB = \left[\begin{array}{cc}-22 & -6 \ 18 & 36\end{array}\right]\]

Key Concepts

Matrix Compatibility

When it comes to matrix multiplication, not all matrices can be multiplied together. They must be compatible. This means that the number of columns in the first matrix must equal the number of rows in the second matrix. For instance, matrix \(C\) in this exercise is a \(2 \times 3\) matrix, while matrix \(B\) is a \(3 \times 2\) matrix.
Because the number of columns in matrix \(C\) (3 columns) is the same as the number of rows in matrix \(B\) (3 rows), matrix multiplication can be performed.
Think of it like a puzzle where the pieces (rows and columns) must align perfectly to fit together. If they don't align, you simply can't multiply those matrices and get a correctly formed output.

Resultant Matrix

After confirming that two matrices can be multiplied, it is time to determine the size of the resultant matrix. The resultant matrix from multiplying a \(2 \times 3\) matrix by a \(3 \times 2\) matrix will have dimensions \(2 \times 2\). Hence, the result of multiplying matrices \(C\) and \(B\) will produce a \(2 \times 2\) matrix.
In matrix multiplication, the size of the resultant matrix is determined by:

• The number of rows from the first matrix (\(C\) in this case)
• The number of columns from the second matrix (\(B\) in this example)

Thus, the dimensional formula for our final matrix is derived from these aspects.

Matrix Elements

The elements of the resultant matrix cannot just be guessed; they are calculated methodically from the corresponding elements of the given matrices. Each element of the resultant matrix is computed using a sum of products of elements from rows of the first matrix and columns of the second matrix.
For example, to find the element at the first row and first column, \((CB)_{11}\), we follow this computation:

• Multiply each element of the first row of \(C\) by the corresponding element in the first column of \(B\).
• Sum these products to get the final value.Again, this can be seen as combing values from the row and column to produce a unique element for the resultant matrix.

By applying this for each element in the resulting matrix, we meticulously calculate all products:\[ CB = \left[\begin{array}{cc}-22 & -6 \ 18 & 36\end{array}\right] \] This concludes the result of the entire multiplication process.