Problem 29 - College Algebra

Question

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

Step-by-Step Solution

Answer

Matrix \(a = [30]\) and matrix \(b = \left[\begin{array}{l} 1 & 2 & 3 & 4 \\ 2 & 4 & 6 & 8 \\3 & 6 & 9 & 12 \\4 & 8 & 12 & 16\end{array}\right]\)

Step 1: Multiply A with B to find matrix a.

To find matrix a, which is A times B, each element in the only row of A is multiplied with corresponding element in the only column of B and then all those numbers are summed. Hence, \(a = A B = (1*1 + 2*2 + 3*3 + 4*4) = 30 \). Thus, \(a = [30]\).

Step 2: Multiply B with A to find matrix b.

To find matrix b, which is B times A, each element in B is multiplied with every element in A. Hence, \(b = B A = \left[\begin{array}{l} 1*1 & 1*2 & 1*3 & 1*4 \\ 2*1 & 2*2 & 2*3 & 2*4 \\3*1 & 3*2 & 3*3 & 3*4 \\4*1 & 4*2 & 4*3 & 4*4\end{array}\right] = \left[\begin{array}{l} 1 & 2 & 3 & 4 \\ 2 & 4 & 6 & 8 \\3 & 6 & 9 & 12 \\4 & 8 & 12 & 16\end{array}\right]\).

Key Concepts

Matrices

Matrices are fundamental structures in mathematics, especially in linear algebra. They are rectangular arrays of numbers arranged in rows and columns. Understanding matrices is crucial for performing various matrix operations like addition, subtraction, and multiplication.

• Each number in a matrix is called an element.
• The dimensions of a matrix are given by the number of rows and columns. For example, matrix A in the exercise is a 1x4 matrix, meaning it has 1 row and 4 columns.
• Matrices can be of different types: square matrices (same number of rows and columns), rectangular matrices, row matrices (only one row), column matrices (only one column), etc.

Matrices are used not only in pure mathematics but across various fields such as computer graphics, physics, and engineering.

Matrix Operations

Matrix operations involve manipulating matrices to produce new matrices. Some of the basic operations include addition, subtraction, and multiplication.

• Addition & Subtraction: These operations are straightforward. Two matrices of the same dimensions can be added or subtracted by adding or subtracting their corresponding elements.
• Multiplication: This is a bit more complex and involves a specific process. For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix will have dimensions equal to the number of rows in the first matrix and the number of columns in the second matrix.

In the exercise, we performed matrix multiplication to find matrices a and b. When multiplying matrix A (1x4) with B (4x1), we ended up with a new matrix which is a single 1x1 matrix (a). In contrast, multiplying B (4x1) with A (1x4) resulted in a 4x4 matrix (b). Each element of the resulting matrix was calculated by taking the dot product of a row from the first matrix with a column from the second matrix.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They form the basis for defining matrix operations since matrices can represent algebraic expressions that simplify complex calculations.

• When dealing with matrix operations, especially multiplication, algebraic expressions are used to represent and solve parts of the computation.
• In matrix multiplication, the individual calculations performed for each element are like algebraic expressions. For example, in this exercise, computing operations like \(1*1 + 2*2 + 3*3 + 4*4\) in matrix multiplication can be seen as solving a linear equation.

Understanding algebraic expressions helps make sense of the steps involved in operations such as matrix multiplication by breaking them into simpler, more manageable parts. This foundation makes it easier to tackle larger and more complex matrix algebra problems.