Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Find, if possible, \(A B\) and \(B A\). \(A=\left[\begin{array}{lll}-3 & 7 & 2\end{array}\right]\) $$ B=\left[\begin{array}{r} 1 \\ 4 \\ -5 \end{array}\right] $$

Step-by-Step Solution

Answer

\(AB = \begin{bmatrix} 15 \end{bmatrix}\) and \(BA\) is not possible.

Step 1: Determine if Multiplications are Possible

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix \(A\) is \(1 \times 3\) and matrix \(B\) is \(3 \times 1\), so \(AB\) is possible and will result in a \(1 \times 1\) matrix. Conversely, \(BA\) is not possible because matrix B would have 3 rows while Matrix A has only 1 column.

Step 2: Calculate \(AB\)

Perform matrix multiplication \(AB\). Matrix \(A\) has elements \([-3, 7, 2]\) and matrix \(B\) has elements \([1, 4, -5]\).Calculate the product: \(-3 \cdot 1 + 7 \cdot 4 + 2 \cdot (-5)\).

Step 3: Simplify the Expression

Simplify the above expression:\(-3 \times 1 = -3\) \\(7 \times 4 = 28\) \\(2 \times (-5) = -10\) \Add these results together:\(-3 + 28 - 10 = 15\).

Step 4: State the Final Results

The matrix multiplication \(AB\) results in a \(1 \times 1\) matrix, which is \[15\]. The product \(BA\) is not possible.

Key Concepts

Matrix Dimensions

Matrix dimensions are crucial to understand because they dictate whether or not you can successfully multiply two matrices. The dimensions of a matrix are expressed as "rows × columns". For instance, matrix \(A\) in our exercise has dimensions \(1 \times 3\), meaning it has one row and three columns.
Matrix \(B\) has the dimensions \(3 \times 1\), meaning it has three rows and one column.
The key rule to note here is: for matrix multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix.

• In our specific problem, the product \(AB\) is possible because the number of columns in \(A\) (which is 3) matches the number of rows in \(B\) (also 3).
• However, \(BA\) is not viable since \(B\) has 3 rows and matrix \(A\), only having 1 column, does not provide a suitable dimension match.

Matrix Product

The matrix product refers to the result of multiplying two matrices. Performing this operation involves a systematic process. When multiplying matrix \(A\) by matrix \(B\) as in our exercise, what you are actually doing is taking the elements of the rows of \(A\) and columns of \(B\), then calculating their products.
The resulting matrix from our example of \(AB\) is a \(1 \times 1\) matrix because \(A\) is \(1 \times 3\) and \(B\) is \(3 \times 1\). The product's dimensions are determined by the outer dimensions of the matrices being multiplied.

• Since \(AB\) results in a single value matrix, it essentially condenses the row-by-column computation into just one value.
• This value is calculated by multiplying the corresponding elements of the rows of \(A\) and the columns of \(B\).

Matrix Calculation

Matrix calculation often extends beyond simple arithmetic and requires careful attention to each element involved in the product. In our provided exercise, the calculation of \(AB\) entails multiplying the corresponding elements of the two matrices and summing these results.
In step-by-step detail, this is how it unfolds:

• First, multiply the first element of the single row of \(A\), which is \(-3\), with the first element of \(B\), which is \(1\), resulting in \(-3\).
• Next, multiply the second element \(7\) of \(A\) with the second element \(4\) of \(B\), yielding \(28\).
• Then, multiply the third and final element \(2\) of \(A\) with the third element \(-5\) of \(B\), which gives \(-10\).

You then add up all these products: \(-3 + 28 - 10 = 15\). This sum represents the single entry in the resulting \(1 \times 1\) matrix \(AB\), making matrix calculation a straightforward but essential process.