Problem 39 - Precalculus

Question

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

Step-by-Step Solution

Answer

\( G^2 + 5G = \begin{bmatrix} 12 & 20 \\ 30 & 42 \end{bmatrix} \)

Step 1: Find the Matrix Product G^2

First, compute the product of matrix \( G \) with itself to find \( G^2 \). The matrices \( G \) and \( G \) are both 2x2 matrices:\[ G = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \]Compute the product:\[ G^2 = G \times G = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \]Using the matrix multiplication formula, calculate each term:- Element (1,1): \( 1 \times 1 + 2 \times 3 = 1 + 6 = 7 \)- Element (1,2): \( 1 \times 2 + 2 \times 4 = 2 + 8 = 10 \)- Element (2,1): \( 3 \times 1 + 4 \times 3 = 3 + 12 = 15 \)- Element (2,2): \( 3 \times 2 + 4 \times 4 = 6 + 16 = 22 \)Thus, \[ G^2 = \begin{bmatrix} 7 & 10 \ 15 & 22 \end{bmatrix} \]

Step 2: Calculate 5G

Next, calculate the scalar multiplication of the matrix \( G \) by 5:\[ 5G = 5 \times \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \]This is achieved by multiplying each element of \( G \) by 5:- Element (1,1): \( 5 \times 1 = 5 \)- Element (1,2): \( 5 \times 2 = 10 \)- Element (2,1): \( 5 \times 3 = 15 \)- Element (2,2): \( 5 \times 4 = 20 \)Thus, \[ 5G = \begin{bmatrix} 5 & 10 \ 15 & 20 \end{bmatrix} \]

Step 3: Add G^2 and 5G

Finally, add the matrices \( G^2 \) and \( 5G \) together:\[ G^2 + 5G = \begin{bmatrix} 7 & 10 \ 15 & 22 \end{bmatrix} + \begin{bmatrix} 5 & 10 \ 15 & 20 \end{bmatrix} \]Add corresponding elements of the matrices:- Element (1,1): \( 7 + 5 = 12 \)- Element (1,2): \( 10 + 10 = 20 \)- Element (2,1): \( 15 + 15 = 30 \)- Element (2,2): \( 22 + 20 = 42 \)The resulting matrix is:\[ G^2 + 5G = \begin{bmatrix} 12 & 20 \ 30 & 42 \end{bmatrix} \]

Key Concepts

Matrix Multiplication

Matrix multiplication allows us to combine two matrices to form a new matrix. It's essential to remember that not all matrices can be multiplied. The rule for matrix multiplication is that the number of columns in the first matrix must equal the number of rows in the second matrix. For instance, a matrix size of 2x3 can be multiplied by a matrix of size 3x2, but not by a 2x2.

Matrix multiplication involves an element-by-element operation across the rows of the first matrix and the columns of the second matrix. Each element in the resultant matrix is the sum of products of corresponding elements. For example, with matrix \( G \), a 2x2 matrix from our exercise:

• To get the element at position (1,1), multiply the first row of \( G \) by the first column of the second \( G \), and sum the results.
• Continue this process for each position in the resultant matrix.

The calculations can become complex but by following the steps methodically, the right result is attainable. Using the step-by-step method enhances accuracy.

Scalar Multiplication

Scalar multiplication is straightforward, where each element in a matrix is multiplied by a scalar value, a single number. For example, if we take matrix \( G \) from the exercise and multiply it by 5, we deal with each element separately.

Follow these simple steps:

• Multiply each entry, be it in row or column, by the scalar. For instance, if \( G \) is multiplied by 5, the element at position (1,1) is 5 times the original element.
• This operation affects all elements uniformly, keeping the relative structure of the original matrix unchanged but altering its scale by the scalar.

This process showcases the beautiful symmetry of scalar multiplication, as it uniformly adjusts the entire matrix without changing its dimensions or shape.

Matrix Addition

Matrix addition is an operation where two matrices of identical dimensions are added together. The corresponding elements of each matrix are summed to form a new matrix. It's crucial to remember that only matrices of the same size can be added. For instance, a 2x3 matrix cannot be added to a 3x2 matrix.

During the addition process:

• Add the elements of the first matrix directly with the corresponding elements of the second matrix.
• For example, if adding \( G^2 \) and 5\( G \), you would sum each matching position element to produce a result at each corresponding position in the new matrix.

This operation is straightforward and forms the basis of many matrix applications that require aggregation of information across datasets.