Problem 29 - Precalculus

Question

perform the indicated operations for each expression, if possible. $$A=\left[\begin{array}{rrr}-1 & 3 & 0 \\\2 & 4 & 1\end{array}\right] \quad B=\left[\begin{array}{rr}0 & 2 & 1 \\\3 & -2 & 4\end{array}\right] \quad C=\left[\begin{array}{rr}0 & 1 \\\2 & -1 \\\3 & 1 \end{array}\right] \quad D=\left[\begin{array}{rr}2 & -3 \\\0 & 1 \\\4 & -2 \end{array}\right]$$ $$E=\left[\begin{array}{rrr}-1 & 0 & 1 \\\2 & 1 & 4 \\\\-3 & 1 & 5 \end{array}\right] \quad F=\left[\begin{array}{r}1 \\\0 \\\\-1\end{array}\right] \quad G=\left[\begin{array}{ll}1 & 2 \\\3 & 4\end{array}\right]$$ $$DG$$

Step-by-Step Solution

Answer

The resulting matrix \( DG \) is \( \begin{bmatrix} -7 & -8 \\ 3 & 4 \\ -2 & 0 \end{bmatrix} \).

Step 1: Verify Dimensions for Matrix Multiplication

Matrix multiplication is only possible when the number of columns in the first matrix matches the number of rows in the second matrix. Matrix \( D \) has dimensions \( 3 \times 2 \), and matrix \( G \) has dimensions \( 2 \times 2 \). Since the number of columns in \( D \) (2) matches the number of rows in \( G \) (2), the multiplication is possible, and the resulting matrix will have dimensions \( 3 \times 2 \).

Step 2: Perform Matrix Multiplication

To multiply matrices \( D \) and \( G \), compute each element by taking the dot product of the corresponding row of \( D \) with the column of \( G \). The resulting matrix \( DG \) is calculated as follows:First row: - \((2 \times 1) + (-3 \times 3) = 2 - 9 = -7\)- \((2 \times 2) + (-3 \times 4) = 4 - 12 = -8\)Second row:- \((0 \times 1) + (1 \times 3) = 3\)- \((0 \times 2) + (1 \times 4) = 4\)Third row:- \((4 \times 1) + (-2 \times 3) = 4 - 6 = -2\)- \((4 \times 2) + (-2 \times 4) = 8 - 8 = 0\)

Step 3: Write the Resulting Matrix

After computing the elements as described, the resulting matrix \( DG \) is:\[DG = \begin{bmatrix}-7 & -8 \3 & 4 \-2 & 0 \\end{bmatrix}\]

Key Concepts

Matrix Dimensions

When working with matrices, understanding their dimensions is crucial. The dimensions of a matrix are given as 'rows x columns'. For example, a matrix with 3 rows and 2 columns is described as a 3x2 matrix. When we write it in this format, it helps us know the shape and layout of the matrix. This is important because certain operations, like multiplication, depend on the alignment of these dimensions.

Before multiplying matrices, always check their dimensions. If you have matrix \(D\) which is \(3 \times 2\), it means it has 3 rows and 2 columns. If you're trying to multiply it by another matrix, like \(G\), which is \(2 \times 2\), check if the number of columns in \(D\) matches the rows in \(G\). In this case, both have 2, so multiplication is possible.

• Understand dimensions as 'rows x columns'
• Match columns of the first matrix with rows of the second
• Only matrices meeting these criteria can be multiplied

Dot Product

The dot product is a critical step in matrix multiplication. It involves taking two vectors: a row from the first matrix and a column from the second matrix. You multiply each corresponding pair of elements from these two vectors and sum up the results to get a single entry in the resulting matrix.

For instance, to calculate the dot product for the first row of matrix \(D\) and the first column of matrix \(G\), perform the following:

• Multiply the first elements of the row and column: \(2 \times 1 = 2\)
• Multiply the second elements: \((-3) \times 3 = -9\)
• Add these products: \(2 - 9 = -7\)

So, the entry in the first row, first column of the new matrix is \(-7\). Repeat this with other rows and columns to fill the resulting matrix.

Resulting Matrix Dimensions

After confirming that two matrices can indeed be multiplied and completing the calculations, you can determine the dimensions of the resulting matrix. This is a direct consequence of the multiplication rule, where the resulting matrix's dimensions are derived from the outer dimensions of the multiplied matrices.

For example, when multiplying a matrix \(3 \times 2\) with a \(2 \times 2\) matrix, the dimensions of the product matrix will be \(3 \times 2\). This means it will have as many rows as the first matrix and as many columns as the second matrix.

To generalize:

• If you multiply a \(m \times n\) matrix by a \(n \times p\) matrix, the resulting matrix will be \(m \times p\)
• This rule highlights the importance of matched inner dimensions for multiplication
• Always expect the new matrix to conform to these determined dimensions