Problem 33 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

(a) ABE = \( \begin{bmatrix} 13 \\ -7 \end{bmatrix} \) and (b) AHE is not defined.

Step 1: Check Compatibility for Multiplication (a) ABE

First, we need to check if the matrices \( A \), \( B \), and \( E \) can be multiplied together to form the product \( ABE \). - \( A \) is a \( 2 \times 2 \) matrix.- \( B \) is a \( 2 \times 3 \) matrix.- \( E \) is a \( 3 \times 1 \) matrix.For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix.\( AB \) is possible because the number of columns in \( A \) (which is 2) matches the number of rows in \( B \) (which is 2).\( BE \) is possible because the number of columns in \( B \) (3) matches the number of rows in \( E \) (3).Thus, \( (AB)E \) is possible with an output dimension of \( 2 \times 1 \).

Step 2: Calculate AB

To find \( AB \), calculate the matrix product of \( A \) and \( B \):\[ A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix} \]Multiply each row of \( A \) by each column of \( B \):1. First Row of AB: - \((2*3) + (-5*1) = 6 - 5 = 1\) - \((2*\frac{1}{2}) + (-5*-1) = 1 + 5 = 6\) - \((2*5) + (-5*3) = 10 - 15 = -5\)2. Second Row of AB: - \((0*3) + (7*1) = 0 + 7 = 7\) - \((0*\frac{1}{2}) + (7*-1) = 0 - 7 = -7\) - \((0*5) + (7*3) = 0 + 21 = 21\)Thus, \( AB = \begin{bmatrix} 1 & 6 & -5 \ 7 & -7 & 21 \end{bmatrix} \).

Step 3: Calculate ABE

Now, we consider the product \( (AB)E \) using the matrix \( AB = \begin{bmatrix} 1 & 6 & -5 \ 7 & -7 & 21 \end{bmatrix} \) from the previous step and \( E = \begin{bmatrix} 1 \ 2 \ 0 \end{bmatrix} \).Multiply each row of \( AB \) by the column of \( E \):1. First Entry of ABE: - \((1*1) + (6*2) + (-5*0) = 1 + 12 = 13\)2. Second Entry of ABE: - \((7*1) + (-7*2) + (21*0) = 7 - 14 = -7\)Thus, \( ABE = \begin{bmatrix} 13 \ -7 \end{bmatrix} \).

Step 4: Check Compatibility for Multiplication (b) AHE

Next, we check if the matrices \( A \), \( H \), and \( E \) can be multiplied together to form the product \( AHE \).- \( A \) is a \( 2 \times 2 \) matrix.- \( H \) is a \( 2 \times 2 \) matrix.- \( E \) is a \( 3 \times 1 \) matrix.\( AH \) is possible because the number of columns in \( A \) (2) matches the number of rows in \( H \) (2).However, \( HE \) is not possible because the number of columns in \( H \) (2) does not match the number of rows in \( E \) (3). Therefore, the product \( AHE \) is not defined due to size mismatch.

Key Concepts

Matrix Compatibility

When working with matrix multiplication, understanding compatibility is crucial. This compatibility depends on the dimensions of the matrices involved. For a successful multiplication between two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. If this condition is not met, the matrices are incompatible for multiplication and the operation cannot be performed.

In the exercise provided, compatibility checks were necessary for products like \( ABE \) and \( AHE \). In example \( ABE \), matrices \( A \) and \( B \) were compatible because \( A \) had 2 columns and \( B \) had 2 rows. After multiplying \( A \) by \( B \), the resulting matrix \( AB \) had 3 columns, which matched the 3 rows in matrix \( E \), making the final multiplication \((AB)E\) possible. However, in the case of \( AHE \), although \( A \) and \( H \) were compatible and could multiply (because they both have 2 rows and 2 columns respectively), \( H \) and \( E \) were not, due to \( H \) having 2 columns and \( E \) having 3 rows. This mismatch made \( AHE \) impossible.

Matrix Dimensions

Understanding matrix dimensions is key when discussing matrix multiplication. A matrix's dimensions are given by the number of rows and columns it contains, described in the format of "rows × columns."

For instance, matrix \( A \) is a \( 2 \times 2 \) matrix, meaning it has 2 rows and 2 columns. Meanwhile, matrix \( B \) is a \( 2 \times 3 \) matrix with 2 rows and 3 columns. The dimension description helps in determining how matrices align for multiplication.

To multiply matrices, ensure that the number of columns in the first matrix matches the number of rows in the second. This alignment is essential for proceeding with the multiplication, forming inner dimensions that should cancel out to enable the matrix operation. The outer dimensions (i.e., the rows of the first matrix and the columns of the second matrix) become the dimensions of the resulting matrix.

Remembering these dimensional rules will not only aid in identifying compatibility but will also help predict the size of the product matrix, offering insight even before computations begin.

Algebraic Operations

Matrix multiplication is one of the core algebraic operations in linear algebra. Unlike simple arithmetic multiplication, matrices adopt specific rules that require strict adherence to dimensional compatibility.

To perform matrix multiplication, you take the dot product of rows of the first matrix with columns of the second matrix. This involves multiplying each element of a row with the corresponding element of a column and then summing these products.

For example, when calculating the product \( AB \) from matrices \( A \) and \( B \), you multiply corresponding elements from the row of \( A \) with the column of \( B \) and sum the results to form each element of the new matrix. This operation is repeated across each row in \( A \) and each column in \( B \) until the matrix \( AB \) is complete. This requires careful sequencing and calculation to ensure every element is properly assessed.

This operation is non-commutative, meaning changing the order of the matrices will likely yield a different result or may not be possible if dimensions do not align. Thus, being meticulous with order and steps in matrix algebra ensures accurate outcomes.