Question

In each of the following cases, find all possible products \(A^{2}, A B, A C,\) and so on a. \(A=\left[\begin{array}{rrr}1 & 2 & 3 \\ -1 & 0 & 0\end{array}\right], B=\left[\begin{array}{rr}1 & -2 \\ \frac{1}{2} & 3\end{array}\right]\) \(C=\left[\begin{array}{rr}-1 & 0 \\ 2 & 5 \\ 0 & 3\end{array}\right]\) b. \(A=\left[\begin{array}{rrr}1 & 2 & 4 \\ 0 & 1 & -1\end{array}\right], B=\left[\begin{array}{rr}-1 & 6 \\ 1 & 0\end{array}\right]\) \(C=\left[\begin{array}{ll}2 & 0 \\ -1 & 1 \\ 1 & 2\end{array}\right]\)

Short answer

In case a, only AC is valid, giving \(\begin{pmatrix} 3 & 19 \\ 1 & 0 \end{pmatrix}\). In case b, only AC works, yielding \(\begin{pmatrix} 4 & 10 \\ -2 & -1 \end{pmatrix}\).

Step-by-step solution

Identify Dimensions for Compatibility

Determine the dimensions of each matrix.For **a:**- Matrix \(A\) is \(2 \times 3\)- Matrix \(B\) is \(2 \times 2\)- Matrix \(C\) is \(3 \times 2\)For **b:**- Matrix \(A\) is \(2 \times 3\)- Matrix \(B\) is \(2 \times 2\)- Matrix \(C\) is \(3 \times 2\)Observe that to multiply two matrices, the number of columns in the first must match the number of rows in the second.

Compute Products for Case a

Evaluate each possible matrix multiplication for case **a:**1. **\(A^2\):** Not defined because the number of columns in \(A\) (3) is not equal to the number of rows in \(A\) (2).2. **\(AB\):** Not defined because the number of columns in \(A\) (3) is not equal to the number of rows in \(B\) (2).3. **\(AC\):** Multiply since \(A\) is \(2 \times 3\) and \(C\) is \(3 \times 2\). \[ A \cdot C = \begin{pmatrix} 1 & 2 & 3 \ -1 & 0 & 0 \end{pmatrix} \begin{pmatrix} -1 & 0 \ 2 & 5 \ 0 & 3 \end{pmatrix} = \begin{pmatrix} (1)(-1) + (2)(2) + (3)(0) & (1)(0) + (2)(5) + (3)(3) \ (-1)(-1) + (0)(2) + (0)(0) & (-1)(0) + (0)(5) + (0)(3) \end{pmatrix} \] Results in: \[ \begin{pmatrix} 3 & 19 \ 1 & 0 \end{pmatrix} \]4. **\(BC\):** Multiply since \(B\) is \(2 \times 2\) and \(C\) is \(3 \times 2\). \[ B \cdot C = \begin{pmatrix} 1 & -2 \ \frac{1}{2} & 3 \end{pmatrix} \begin{pmatrix} -1 & 0 \ 2 & 5 \end{pmatrix} \] Results not computable, dimensions mismatch (2x3 output expected from two 2x2 in part **b**).

Compute Products for Case b

Evaluate the possible matrix multiplications for case **b:**1. **\(A^2\):** Not defined because the number of columns in \(A\) (3) is not equal to the number of rows in \(A\) (2).2. **\(AB\):** Multiplication not possible (because A (2x3) cannot be multiplied with B (2x2)). 3. **\(AC\):** Multiply since \(A\) is \(2 \times 3\) and \(C\) is \(3 \times 2\). \[ A \cdot C = \begin{pmatrix} 1 & 2 & 4 \ 0 & 1 & -1 \end{pmatrix} \begin{pmatrix} 2 & 0 \ -1 & 1 \ 1 & 2 \end{pmatrix} \] Results in: \[ \begin{pmatrix} 4 & 10 \ -2 & -1 \end{pmatrix} \]4. **\(BC\):** Multiply because \(B\) is \(2 \times 2\) and \(C\) is \(2 \times 3\). \[ B \cdot C = \begin{pmatrix} -1 & 6 \ 1 & 0 \end{pmatrix} \begin{pmatrix} 2 & 0 \ -1 & 1 \ 1 & 2 \end{pmatrix} \] Results not computable, dimensions mismatch (2x3 output expected from two 2x2 in part **b**).

Summarize Results

In case **a**, only the product \(AC\) is possible, resulting in the matrix \(\begin{pmatrix} 3 & 19 \ 1 & 0 \end{pmatrix}\). In case **b**, the product \(AC\) is possible, resulting in the matrix \(\begin{pmatrix} 4 & 10 \ -2 & -1 \end{pmatrix}\). The other combinations in both cases result in undefined products due to dimension disparities.

Key concepts

Matrix Dimensions

Matrix dimensions are fundamental when working with matrix multiplication. Each matrix is defined by its number of rows and columns, which are written as "rows x columns."
For instance, in the given exercise, matrix \(A\) has a dimension of \(2 \times 3\) because it has 2 rows and 3 columns.

• Matrix \(B\) is \(2 \times 2\), meaning it has 2 rows and 2 columns.
• Matrix \(C\) is \(3 \times 2\), indicating 3 rows and 2 columns.

These dimensions are essential for determining how matrices interact with one another during multiplication.

Matrix Compatibility

Matrix compatibility is crucial when deciding whether two matrices can be multiplied. A matrix can only be multiplied by another if the number of columns of the first matrix matches the number of rows of the second matrix.
This rule arises from the requirement that each component in a row from the first matrix pairs with each component from a column of the second matrix.
In the exercise, we see examples of such compatibility:

• The matrices \(A\) and \(C\) are compatible for multiplication because \(A\) has 3 columns and \(C\) has 3 rows.
• Meanwhile, \(A^2\) and \(AB\) are not possible due to mismatched dimensions, where the number of columns in the first matrix does not equal the number of rows in itself or the other matrices \((B)\).

Matrix Product Calculation

Once matrix compatibility is established, calculating the matrix product is the next step. To find the product, you multiply and sum the corresponding components from rows and columns.
Let's see how this works using the example of multiplying matrices \(A\) and \(C\):1. Identify corresponding cells: For instance, the first element in row 1 of the product is calculated as follows: \((1)(-1) + (2)(2) + (3)(0)\), which gives 3.
2. Continue similarly for other elements.When multiplying \(A\) and \(C\), the calculation results in:\[A \cdot C = \begin{pmatrix} 1 & 2 & 3 \ -1 & 0 & 0 \end{pmatrix} \begin{pmatrix} -1 & 0 \ 2 & 5 \ 0 & 3 \end{pmatrix} = \begin{pmatrix} 3 & 19 \ 1 & 0 \end{pmatrix}.\]This product exemplifies how multiplication leverages the dimensions and compatibility to yield a new matrix with new dimensions, specifically \(2 \times 2\) in this case.