Problem 32 - 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. $$ \begin{array}{ll}{\text { (a) }(D A) B} & {\text { (b) } D(A B)}\end{array} $$

Step-by-Step Solution

Answer

Both operations are valid: (a) \((DA)B = \begin{bmatrix} 48 & 1 & 88 \end{bmatrix}\); (b) \(D(AB) = \begin{bmatrix} 28 & 5.5 & 28 \end{bmatrix}\).

Step 1: Calculate Matrix Product DA

Multiply matrices \(D\) and \(A\). Matrix \(D\) is a \(1 \times 2\) matrix and \(A\) is a \(2 \times 2\) matrix, which makes the product \(DA\) possible. \[ D = \begin{bmatrix} 7 & 3 \end{bmatrix}, \quad A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \]Matrix multiplication is performed by calculating the dot product of rows and columns.\[ DA = \begin{bmatrix} 7 \times 2 + 3 \times 0 & 7 \times (-5) + 3 \times 7 \end{bmatrix} = \begin{bmatrix} 14 & 6 \end{bmatrix} \]

Step 2: Assess Product (D A) B

Now we check if the product \((DA)B\) is possible. Matrix \(DA\) is \(1 \times 2\) and matrix \(B\) is \(2 \times 3\). Since the number of columns in \(DA\) is equal to the number of rows in \(B\), the multiplication \((DA)B\) can be performed. \( B = \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix} \)We now calculate \((DA)B\):\[ (DA)B = \begin{bmatrix} 14 & 6 \end{bmatrix} \times \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix} = \begin{bmatrix} 14 \times 3 + 6 \times 1 & 14 \times \frac{1}{2} + 6 \times (-1) & 14 \times 5 + 6 \times 3 \end{bmatrix} = \begin{bmatrix} 48 & 1 & 88 \end{bmatrix} \]

Step 3: Calculate Matrix Product AB

Next, let's calculate matrix product \(AB\). Matrix \(A\) is \(2 \times 2\) and matrix \(B\) is \(2 \times 3\), so the product \(AB\) is valid and will result in a \(2 \times 3\) matrix. \[ A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{bmatrix} \]Multiply \(A\) and \(B\) by multiplying rows of \(A\) by columns of \(B\) and summing up the products.\[ AB = \begin{bmatrix} 2 \times 3 + (-5) \times 1 & 2 \times \frac{1}{2} + (-5) \times (-1) & 2 \times 5 + (-5) \times 3 \ 0 \times 3 + 7 \times 1 & 0 \times \frac{1}{2} + 7 \times (-1) & 0 \times 5 + 7 \times 3 \end{bmatrix} = \begin{bmatrix} 1 & 5.5 & -5 \ 7 & -7 & 21 \end{bmatrix} \]

Step 4: Assess Product D(AB)

Determine if it is possible to multiply \(D\) with \(AB\). Matrix \(D\) is \(1 \times 2\) and matrix \(AB\) is \(2 \times 3\), which allows for the product \(D(AB)\). Proceed with the multiplication:\[ D(AB) = \begin{bmatrix} 7 & 3 \end{bmatrix} \times \begin{bmatrix} 1 & 5.5 & -5 \ 7 & -7 & 21 \end{bmatrix} = \begin{bmatrix} 7 \times 1 + 3 \times 7 & 7 \times 5.5 + 3 \times (-7) & 7 \times (-5) + 3 \times 21 \end{bmatrix} = \begin{bmatrix} 28 & 5.5 & 28 \end{bmatrix} \]

Key Concepts

Matrix Dimensions

When dealing with matrices, understanding their dimensions is crucial. A matrix's dimension is defined by the number of rows and columns it possesses, expressed as ‘rows by columns’ in the format. For example, matrix \( A \) in the exercise is a \( 2 \times 2 \) matrix, while matrix \( B \) is a \( 2 \times 3 \) matrix.

To understand how matrix multiplication works, you need to determine if the product is possible based on these dimensions. In any matrix multiplication, the number of columns in the first matrix must match the number of rows in the second matrix. If this condition is not met, the matrices cannot be multiplied.

For the exercise with matrices \( D \) and \( A \), \( D \) has a dimension of \( 1 \times 2 \) and \( A \) is \( 2 \times 2 \), meaning their multiplication is valid. This compatibility between dimensions allows the computation of their product.

• Check dimensions: Make sure the number of columns of the first matrix equals the number of rows of the second matrix.
• Resulting dimension: The resulting matrix from the multiplication will have dimensions equal to the number of rows of the first matrix and the number of columns of the second matrix.

Dot Product

In matrix multiplication, calculating the dot product is key. It involves multiplying corresponding entries from a row of the first matrix and a column of the second matrix, then summing those products.
The dot product method simplifies the multiplication of matrices, as we saw with \( D \) and \( A \): you multiply elements across each row and column pair, then add those products to get each entry in the resulting matrix.

Here's how it works:

• Select a row from the first matrix and a column from the second matrix.
• Multiply each element of the row with the corresponding element of the column.
• Sum the results of these multiplications to get the entry of the resulting matrix.

This process is repeated for all row-column combinations to fill the entire product matrix. For instance, in our example, the first entry of the product matrix \( DA \) was computed as \( 7 \times 2 + 3 \times 0 = 14 \). Each step is straightforward, so take your time to ensure accuracy.

Matrix Product Validation

After performing any matrix multiplication, it's vital to validate the results and ensure the operations were correctly executed. Matrix product validation involves verifying that the conditions for multiplication were properly met and that calculations were performed accurately.

For example, when calculating \((DA)B\) in the exercise, ensure matrices \( DA \) and \( B \) are dimensionally compatible for multiplication. \( DA \) ended up being a \( 1 \times 2 \) matrix, and you confirmed \( B \) as a \( 2 \times 3 \) matrix. Approval of such matrix validation is essential to perform the actual multiplication operation successfully.

• Check if the matrices are compatible for multiplication (column count of first matrix matches row count of second matrix).
• Verify each entry of the resulting matrix to ensure calculations were done correctly.
• Ensure the final dimensions of your product align with expectations (from our initial matrix dimensions).

Thus, ensuring the multiplication's feasibility and correctness helps avoid errors in complex matrix operations.