Problem 33 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

(a) \(\begin{bmatrix} 13 \\ -7 \end{bmatrix}\); (b) Not possible.

Step 1: Determine Matrix Dimensions

First, find the dimensions of matrices involved in the operations to ensure the multiplication is possible.For (a), we will multiply matrices \(A, B,\) and \(E\) where \(A\) is \(2 \times 2\), \(B\) is \(2 \times 3\), and \(E\) is \(3 \times 1\).For (b), matrices \(A\) and \(H\) are \(2 \times 2\) and \(2 \times 2\) respectively, and \(E\) remains \(3 \times 1\).

Step 2: Check Compatibility for Multiplication (a)

For \(A B E\), check that the number of columns in \(A\) (2) matches the number of rows in \(B\) (2), which it does, making \(AB\) a \(2 \times 3\) matrix. Then, check that the number of columns in \(AB\) (3) matches the number of rows in \(E\) (3), which it does, so \(ABE\) is possible resulting in a \(2 \times 1\) matrix.

Step 3: Perform Multiplication (a)

First calculate \(AB\):\[ AB = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix} = \begin{bmatrix} (2*3) + (-5*1) & (2*\frac{1}{2}) + (-5 * -1) & (2*5) + (-5*3) \ (0*3) + (7*1) & (0*\frac{1}{2}) + (7*-1) & (0*5) + (7*3) \end{bmatrix} \] which simplifies to \[ AB = \begin{bmatrix} 6 - 5 & 1 + 5 & 10 - 15 \ 7 + 0 & 0 - 7 & 0 + 21 \end{bmatrix} = \begin{bmatrix} 1 & 6 & -5 \ 7 & -7 & 21 \end{bmatrix} \].Next, calculate \((AB)E\):\[ \begin{bmatrix} 1 & 6 & -5 \ 7 & -7 & 21 \end{bmatrix}\begin{bmatrix} 1 \ 2 \ 0 \end{bmatrix} = \begin{bmatrix} (1*1) + (6*2) + (-5*0) \ (7*1) + (-7*2) + (21*0) \end{bmatrix} \] This simplifies to \[ \begin{bmatrix} 1 + 12 + 0 \ 7 - 14 + 0 \end{bmatrix} = \begin{bmatrix} 13 \ -7 \end{bmatrix} \].

Step 4: Check Compatibility for Multiplication (b)

For \(AH E\), check that the number of columns in \(A\) (2) matches the number of rows in \(H\) (2), which it does. The resultant \(AH\) will be a \(2 \times 2\) matrix. However, the number of columns in \(AH\) (2) does not match the number of rows in \(E\) (3), meaning \(AHE\) is not possible.

Step 5: Conclusion

(a) Matrix multiplication \(ABE\) results in \(\begin{bmatrix} 13 \ -7 \end{bmatrix}\).(b) Matrix multiplication \(AHE\) is not possible due to dimension mismatch.

Key Concepts

Matrix Algebra

Matrix Algebra is the branch of mathematics that deals with matrices and their operations. It involves performing various mathematical operations on matrices such as addition, subtraction, and multiplication. Matrix multiplication is one of the core operations frequently utilized in mathematical computations, computer graphics, and engineering applications. In the context of the given exercise, we focus primarily on matrix multiplication to solve the problem.Understanding how to perform matrix multiplication is crucial:

• Each element of the resulting matrix is a dot product of rows and columns from the matrices being multiplied.
• The operation is associative, meaning (\(A(BC) = (AB)C\)). However, note that it is not commutative; in other words, (\(AB eq BA\)).

The compatibility of dimensions plays a vital role in the possibility and outcome of the multiplication operation. This exercise emphasizes the need to carefully examine dimensions before performing the multiplication.

Matrix Dimensions

Matrix Dimensions are defined by the number of rows and columns a matrix possesses. Understanding these dimensions is crucial to performing matrix operations successfully.Consider the dimensions of matrices involved in exercises:- Matrix \(A\) is a \(2 \times 2\) matrix- Matrix \(B\) is a \(2 \times 3\) matrix- Matrix \(E\) is a \(3 \times 1\) matrixIn matrix multiplication, the output matrix's dimensions depend on the dimensions of the two matrices being multiplied. For example, multiplying a \(2 \times 2\) matrix with a \(2 \times 3\) matrix results in a \(2 \times 3\) matrix, provided the compatibility condition is met.Understanding dimensions helps anticipate the shape of the resulting matrix, enabling prediction and troubleshooting of operations, such as in our step by step solution from the exercise.

Matrix Compatibility

Matrix Compatibility determines whether it is possible to perform operations between matrices. To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix.In the exercise:

• For expression \(AB\), matrix \(A\) has \(2\) columns and matrix \(B\) has \(2\) rows. Therefore, they are compatible for multiplication, resulting in a \(2 \times 3\) matrix.
• For expression \(BE\), the resultant \(AB\) was found to have \(3\) columns, and matrix \(E\) has \(3\) rows. They are compatible, leading to a \(2 \times 1\) final matrix.
• For \(AH\), both are \(2 \times 2\) matrices and multiplication is possible, but adding matrix \(E\) to this sequence as \(AHE\) fails because the resulting \(AH\) matrix needs \(2\) columns to meet \(E\)'s \(3\) rows.

Ensuring matrices' dimensions satisfy compatibility conditions is key to successfully performing matrix multiplication.