Problem 25 - 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) \(A D\) (b) \(D A\)

Step-by-Step Solution

Answer

(a) AD cannot be performed. (b) DA = \( \begin{bmatrix} 14 & -14 \end{bmatrix} \)

Step 1: Check compatibility for multiplication (a) AD

To multiply matrices, the number of columns in the first matrix must match the number of rows in the second matrix. Here, A is a 2x2 matrix, and D is a 1x2 matrix. Since the number of columns in A (which is 2) does not match the number of rows in D (which is 1), the multiplication AD cannot be performed.

Step 2: Check compatibility for multiplication (b) DA

Now, consider the multiplication DA. Matrix D is 1x2 and matrix A is 2x2. The number of columns in D (which is 2) matches the number of rows in A (which is also 2), so multiplication is possible. The resulting matrix will have dimensions 1x2.

Step 3: Perform the multiplication DA

Matrix D is \( \begin{bmatrix} 7 & 3 \end{bmatrix} \) and matrix A is \( \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \). Multiply each row of D by each column of A:For the first element: \(7 \cdot 2 + 3 \cdot 0 = 14 \)For the second element: \(7 \cdot (-5) + 3 \cdot 7 = -35 + 21 = -14 \)The resulting matrix DA is: \( \begin{bmatrix} 14 & -14 \end{bmatrix} \).

Key Concepts

Matrix Compatibility

When dealing with matrix multiplication, one of the most important aspects to consider is compatibility. If you compare this to putting puzzle pieces together, they must fit together perfectly. In the context of matrices, this means the number of columns in the first matrix must match the number of rows in the second matrix.

If this requirement is not met, the operation cannot be performed. For instance, in the problem above, when attempting to multiply matrix \( A \) by matrix \( D \) (expressed as \( AD \)), the operation is not possible.

• Matrix \( A \) has 2 columns.
• Matrix \( D \) has 1 row.

Since 2 does not equal 1, matrix \( A \) and \( D \) are not compatible for this order of multiplication.

Matrix Dimensions

Understanding the dimensions of matrices helps in determining whether matrices can be multiplied. The dimensions are expressed as \( m \times n \), with \( m \) representing the number of rows and \( n \) indicating the number of columns.

In our example:

• Matrix \( A \) has dimensions \( 2 \times 2 \).
• Matrix \( D \) is \( 1 \times 2 \).
• Matrix \( H \) is a \( 2 \times 2 \) matrix.

The dimensions directly impact the resulting matrix's dimensions, which in turn affects whether or not a multiplication operation can be legally performed.

When multiplying \( D \) by \( A \) (i.e., \( DA \)), the compatible multiplication produces a matrix with dimensions equal to the outer dimensions, \( 1 \times 2 \).

Algebraic Operations

Matrix multiplication is a core algebraic operation used extensively in various fields like engineering and computer science. Unlike element-wise operations, this multiplication couples elements from rows and columns rather than just corresponding positions.

Given matrices \( D \) and \( A \), the multiplication \( DA \) is calculated as follows:

• Multiply each element of the sole row in \( D \) by each corresponding element in the columns of \( A \).
• The first element of the resulting matrix is obtained by multiplying values: \( 7 \times 2 + 3 \times 0 = 14 \).
• For the second element: \( 7 \times (-5) + 3 \times 7 = -35 + 21 = -14 \).

Thus, the resulting matrix from \( DA \) is \( \begin{bmatrix} 14 & -14 \end{bmatrix} \), showcasing how the layout of elements changes through multiplication.