Problem 31 - Precalculus: Mathematical for Calculus

Question

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as $$\begin{array}{l} A=\left[\begin{array}{rr} 2 & -5 \\ 0 & 7 \end{array}\right] \quad B=\left[\begin{array}{rrrr} 3 & \frac{1}{2} & 5 \\ 1 & -1 & 3 \end{array}\right] \quad C=\left[\begin{array}{rrr} 2 & -\frac{5}{2} & 0 \\ 0 & 2 & -3 \end{array}\right] \\ D=\left[\begin{array}{rrr} 7 & 3 \end{array}\right] & E=\left[\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right] \\ F=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \quad G=\left[\begin{array}{rrr} 5 & -3 & 10 \\ 6 & 1 & 0 \\ -5 & 2 & 2 \end{array}\right] \end{array}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$G E$$

Step-by-Step Solution

Answer

The product \(GE\) is \( \begin{bmatrix} -1 \\ 8 \\ -1 \end{bmatrix} \).

Step 1: Check Matrix Compatibility for Multiplication

Matrix multiplication is possible when the number of columns in the first matrix is equal to the number of rows in the second matrix. Matrix \(G\) is a 3x3 matrix, and matrix \(E\) is a 3x1 matrix. Since the number of columns in \(G\) (3) is equal to the number of rows in \(E\) (3), the multiplication \(GE\) is possible.

Step 2: Multiply the Matrices

To multiply \(G\) and \(E\), compute the dot product of each row of \(G\) with the column of \(E\). The resulting product will be a 3x1 matrix.\[ G \cdot E = \begin{bmatrix} 5 & -3 & 10 \ 6 & 1 & 0 \ -5 & 2 & 2 \end{bmatrix} \begin{bmatrix} 1 \ 2 \ 0 \end{bmatrix} = \begin{bmatrix} (5 \times 1) + (-3 \times 2) + (10 \times 0) \ (6 \times 1) + (1 \times 2) + (0 \times 0) \ (-5 \times 1) + (2 \times 2) + (2 \times 0) \end{bmatrix} = \begin{bmatrix} 5 - 6 \ 6 + 2 \ -5 + 4 \end{bmatrix} \]This simplifies to \[ \begin{bmatrix} -1 \ 8 \ -1 \end{bmatrix} \].

Step 3: Write the Result

The product of matrices \(G\) and \(E\) is the 3x1 matrix \( \begin{bmatrix} -1 \ 8 \ -1 \end{bmatrix} \).

Key Concepts

Dot Product

The dot product is a foundational concept in matrix multiplication and involves multiplying corresponding elements of a row and a column, then summing these products. For matrix multiplication, the dot product helps compute the entries of the resulting matrix.

Imagine you have two matrices, the first being a 3x3 matrix and the second a 3x1 matrix. When multiplying these, you'll focus on one row of the first matrix (matrix G) and the single column of the second matrix (matrix E).

To find each element in the resulting matrix, do the following for each row of the first matrix:

• Multiply the first element of the row by the first element of the column.
• Multiply the second element of the row by the second element of the column.
• Multiply the third element of the row by the third element of the column.
• Add these products together to get the corresponding element in the resulting matrix.

The dot product is repeated for each row of the first matrix to fill the resulting matrix. This is how you get a new matrix from the product of two matrices. It’s a step-by-step process but very straightforward once you get the hang of it!

Matrix Compatibility

Matrix compatibility is crucial for matrix multiplication. Not all pairs of matrices can be multiplied. The defining rule is simple: the number of columns in the first matrix must match the number of rows in the second matrix.

Consider matrix G, which is a 3x3 matrix, and matrix E, which is a 3x1 matrix. To determine if the multiplication \(GE\) is possible, compare the sizes of these matrices. Matrix G has 3 columns, and matrix E has 3 rows. Because these numbers match, the multiplication is valid.

When matrices are compatible, the resulting matrix takes a new shape: the number of rows from the first matrix (G) and the number of columns from the second matrix (E). Thus, in this case, multiplication yields a 3x1 matrix. Always keep in mind this compatibility rule before attempting to multiply two matrices.

3x1 Matrix Multiplication

3x1 matrix multiplication involves multiplying a larger matrix with three columns by a smaller matrix with a single column. Here, we specifically look at multiplying a 3x3 matrix by a 3x1 matrix.

In the exercise, matrix G is a 3x3 matrix and matrix E is a 3x1 matrix. When you multiply these, you follow the rule of getting a row-by-column product. The resulting product is another 3x1 matrix.

The process is quite methodical:

• Start with the first row of matrix G and the column in matrix E.
• Compute the dot product for this row and column to get the first element of the new matrix.
• Repeat the process for the second row of matrix G and the column in matrix E to get the second element.
• Finally, do the same for the third row to get the third and final element of the resulting matrix.

This approach ensures that each component of the new matrix is accurately calculated, matching dimensions as per matrix multiplication rules.