Problem 31 - College Algebra

Question

Find (if possible) the following matrices: \(a, A B\) \(\boldsymbol{b}, B A\) $$A=\left[\begin{array}{rrr}1 & -1 & 4 \\\4 & -1 & 3 \\\2 & 0 & -2\end{array}\right], \quad B=\left[\begin{array}{rrr}1 & 1 & 0 \\\1 & 2 & 4 \\\1 & -1 & 3\end{array}\right]$$

Step-by-Step Solution

Answer

Matrix \(a = AB = \left[ \begin{array}{ccc} 6 & -2 & 12 \ 5 & 6 & 12 \ 0 & 2 & 8 \end{array} \right]\) and Matrix \(b = BA = \left[ \begin{array}{ccc} 5 & -2 & 7 \ 21 & -2 & 4 \ 8 & -1 & 7 \end{array} \right]\)

Step 1 - Matrix Multiplication Definition

Recall that, in order to multiply two matrices together, the number of columns in the first matrix must equal the number of rows in the second matrix. Thus, both AB and BA are possible because A and B are square matrices of size 3x3. The definition is: \( (AB)_{ij} = \sum_{k=1}^n a_{ik}b_{kj} \).

Step 2 - Calculate AB

Calculate the product AB. For each entry in the resulting matrix, multiply the corresponding row from the A matrix by the corresponding column from the B matrix, and sum the results. So, the resulting matrix \(a\) will be: \[ a = A B = \left[ \begin{array}{ccc} 6 & -2 & 12 \ 5 & 6 & 12 \ 0 & 2 & 8 \end{array} \right] \].

Step 3 - Calculate BA

Calculate the product BA. Again, for each entry in the resulting matrix, multiply the corresponding row from the B matrix by the corresponding column from the A matrix, and sum the results. So, the resulting matrix \(b\) will be: \[ b = B A = \left[ \begin{array}{ccc} 5 & -2 & 7 \ 21 & -2 & 4 \ 8 & -1 & 7 \end{array} \right] \].

Key Concepts

square matrices

A square matrix is a type of matrix where the number of rows is equal to the number of columns. This characteristic represents an essential feature in the world of matrices as it allows for certain mathematical operations, such as the calculation of a determinant and the ability to multiply two matrices in both orders, like what we see in our exercise with matrices A and B.
A key point to remember is that all diagonal elements in a square matrix are significant because they can determine properties like singularity or invertibility. Whether we're dealing with a 2x2 or a 3x3 matrix, this definition remains the same. In our example, matrices A and B are both 3x3, making them suitable candidates for matrix multiplication both as \(AB\) and \(BA\).
Recognizing a square matrix is the first step when preparing to multiply matrices, as it confirms that both multiplication directions can produce valid results.

matrix product

Matrix multiplication, also referred to as the matrix product, involves a systematic process where two matrices combine to form another matrix. This can only happen if the rule of inner dimensions alignment is met: columns of the first matrix must equal rows of the second.
In our exercise, both matrices A and B are 3x3, which satisfy this condition, enabling us to find the products \(AB\) and \(BA\). The key to multiplying two matrices is summarizing the product of elements: we take the rows of the first matrix and multiply them by the columns of the second matrix, and then add up these products. It's like using the elements of one matrix to navigate through another!
Thus, calculated results for the matrix product in our exercise show distinct outcomes for \(AB\) and \(BA\), highlighting that the matrix product is not commutative.

matrix rows and columns

Understanding the layout of matrix rows and columns is crucial when multiplying matrices. Rows are the horizontal lines, while columns stand vertically, and their interaction during multiplication decides the resulting elements in a new matrix.
For example, when calculating \( (AB)_{ij} \) as shown in the exercise, the idea is to pick row \( i \) from matrix A and column \( j \) from matrix B for the operation. By taking each element in the chosen row and column, multiplying them together, and summing those products, we find each entry of the resulting product matrix.
This row-column strategy applies to any pair of conformable matrices and must be respected to produce accurate and meaningful matrix products. It's this interplay that drives all matrix multiplication, conveniently packaged in rows and columns.