Question

Given the matrices \(\mathbf{A}\) and \(\mathbf{B}\), find the product \(\mathbf{A B}\). Also, find the product BA in each case in which it is defined. $$ \mathbf{A}=\left(\begin{array}{rr} 3 & 2 \\ -1 & 4 \\ 5 & -2 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rrr} 3 & 0 & 2 \\ -1 & 2 & -3 \end{array}\right) $$

Short answer

The product of matrices \(\mathbf{A}\) and \(\mathbf{B}\) is: $$ \mathbf{A} \cdot\mathbf{B} = \left(\begin{array}{rrr} 7 & 4 & 0 \\ -7 & 8 & -14 \\ 17 & -4 & 16 \end{array}\right) $$ The product of matrices \(\mathbf{B}\) and \(\mathbf{A}\) is: $$ \mathbf{B} \cdot\mathbf{A} = \left(\begin{array}{rr} 4 & 2 \\ -2 & 8 \end{array}\right) $$

Step-by-step solution

Identify dimensions of matrices

Matrix \(\mathbf{A}\) is a 3x2 matrix and matrix \(\mathbf{B}\) is a 2x3 matrix. Since the number of columns in \(\mathbf{A}\) is equal to the number of rows in \(\mathbf{B}\), the product \(\mathbf{A} \cdot\mathbf{B}\) is defined. Now let's calculate the product.

Multiplication of Matrices A and B

To find the product of two matrices, we take the dot product of each row of the first matrix with each column of the second matrix. In this case \((\mathbf{A} \cdot\mathbf{B})_{ij} = \sum_{k} \mathbf{A}_{ik} \mathbf{B}_{kj}\). $$ \mathbf{A} \cdot\mathbf{B} = \left(\begin{array}{rr} 3 & 2 \\ -1 & 4 \\ 5 & -2 \end{array}\right) \cdot \left(\begin{array}{rrr} 3 & 0 & 2 \\ -1 & 2 & -3 \end{array}\right) $$ Let's find the elements in the product matrix step by step.

Calculate elements of product matrix

Element \((\mathbf{A} \cdot\mathbf{B})_{11} = 3 \cdot 3 + 2 \cdot (-1) = 7\). Element \((\mathbf{A} \cdot\mathbf{B})_{12} = 3 \cdot 0 + 2 \cdot 2 = 4\). Element \((\mathbf{A} \cdot\mathbf{B})_{13} = 3 \cdot 2 + 2 \cdot (-3) = 0\). Element \((\mathbf{A} \cdot\mathbf{B})_{21} = (-1) \cdot 3 + 4 \cdot (-1) = -7\). Element \((\mathbf{A} \cdot\mathbf{B})_{22} = (-1) \cdot 0 + 4 \cdot 2 = 8\). Element \((\mathbf{A} \cdot\mathbf{B})_{23} = (-1) \cdot 2 + 4 \cdot (-3) = -14\). Element \((\mathbf{A} \cdot\mathbf{B})_{31} = 5 \cdot 3 + (-2) \cdot (-1) = 17\). Element \((\mathbf{A} \cdot\mathbf{B})_{32} = 5 \cdot 0 + (-2) \cdot 2 = -4\). Element \((\mathbf{A} \cdot\mathbf{B})_{33} = 5 \cdot 2 + (-2) \cdot (-3) = 16\). Now, let's write the final product matrix.

Write the product A*B

The product matrix \(\mathbf{A} \cdot\mathbf{B}\) looks like this: $$ \mathbf{A} \cdot\mathbf{B} = \left(\begin{array}{rrr} 7 & 4 & 0 \\ -7 & 8 & -14 \\ 17 & -4 & 16 \end{array}\right) $$ Now, let's check if the product \(\mathbf{B} \cdot\mathbf{A}\) is defined.

Check if the product B*A is defined

Matrix \(\mathbf{B}\) is a 2x3 matrix, and matrix \(\mathbf{A}\) is a 3x2 matrix. Since the number of columns in \(\mathbf{B}\) is equal to the number of rows in \(\mathbf{A}\), the product \(\mathbf{B} \cdot\mathbf{A}\) is defined. Now let's calculate the product.

Multiplication of Matrices B and A

To find the product of two matrices, we take the dot product of each row of the first matrix with each column of the second matrix. In this case \((\mathbf{B} \cdot\mathbf{A})_{ij} = \sum_{k} \mathbf{B}_{ik} \mathbf{A}_{kj}\). $$ \mathbf{B} \cdot\mathbf{A} = \left(\begin{array}{rrr} 3 & 0 & 2 \\ -1 & 2 & -3 \end{array}\right) \cdot \left(\begin{array}{rr} 3 & 2 \\ -1 & 4 \\ 5 & -2 \end{array}\right) $$ Let's find the elements in the product matrix step by step.

Calculate elements of product matrix

Element \((\mathbf{B} \cdot\mathbf{A})_{11} = 3 \cdot 3 - 1 \cdot 5 = 4\). Element \((\mathbf{B} \cdot\mathbf{A})_{12} = 3 \cdot 2 - 1 \cdot 4 = 2\). Element \((\mathbf{B} \cdot\mathbf{A})_{21} = 0 \cdot 3 + 2 \cdot (-1) = -2\). Element \((\mathbf{B} \cdot\mathbf{A})_{22} = 0 \cdot 2 + 2 \cdot 4 = 8\). Now, let's write the final product matrix.

Write the product B*A

The product matrix \(\mathbf{B} \cdot\mathbf{A}\) looks like this: $$ \mathbf{B} \cdot\mathbf{A} = \left(\begin{array}{rr} 4 & 2 \\ -2 & 8 \end{array}\right) $$ In conclusion, matrix product of matrices \(\mathbf{A}\) and \(\mathbf{B}\) in both orders are defined and can be found as: $$ \mathbf{A} \cdot\mathbf{B} = \left(\begin{array}{rrr} 7 & 4 & 0 \\ -7 & 8 & -14 \\ 17 & -4 & 16 \end{array}\right) $$ and $$ \mathbf{B} \cdot\mathbf{A} = \left(\begin{array}{rr} 4 & 2 \\ -2 & 8 \end{array}\right) $$

Key concepts

Matrices

A matrix is a rectangular array of numbers organized in rows and columns. These numbers are elements of the matrix. Understanding matrices is essential in various fields such as computer science, physics, and engineering because they can represent data or systems of equations effectively. For example, in our problem, we have matrices \( \mathbf{A} \) and \( \mathbf{B} \). Matrix \( \mathbf{A} \) is a 3x2 matrix, which means it has 3 rows and 2 columns. On the other hand, matrix \( \mathbf{B} \) is a 2x3 matrix with 2 rows and 3 columns. Both matrices can hold information that we can use for matrix operations, such as addition or multiplication.

Dot Product

In matrix operations, the dot product plays a critical role, especially in matrix multiplication. To understand the dot product, we must first learn about vectors. A vector is a one-dimensional array of numbers. The dot product is defined for two vectors of the same dimension, calculated by multiplying the corresponding elements and then summing the results.

• For example, for vectors \( \mathbf{u} = [u_1, u_2] \) and \( \mathbf{v} = [v_1, v_2] \), the dot product \( \mathbf{u} \cdot \mathbf{v} = u_1 \times v_1 + u_2 \times v_2 \).

In the context of matrices, this concept extends to rows and columns; to multiply matrices, we calculate the dot product of rows from the first matrix with columns from the second matrix, element by element.

Matrix Dimensions

Matrix dimensions are crucial when performing matrix operations like multiplication. The dimension of a matrix is given by the number of rows and columns it possesses, denoted as 'rows x columns'. This tells you the layout of the matrix, which is foundational for defining operations.

• For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix.
• If matrix \( \mathbf{A} \) is of size 3x2 and matrix \( \mathbf{B} \) of size 2x3, matrix \( \mathbf{A} \cdot \mathbf{B} \) is valid, resulting in a matrix of size 3x3.

However, matrix multiplication is not commutative; \( \mathbf{A} \cdot \mathbf{B} \) does not necessarily equal \( \mathbf{B} \cdot \mathbf{A} \). Thus, knowing dimensions helps to determine feasibility and size of the resulting matrix.

Matrix Product Calculation

Calculating the product of two matrices involves a systematic approach using dot products. Here's how you can do it:

• Identify the dimensions of both matrices to ensure multiplication is possible.
• Follow by aligning each row of the first matrix with each column of the second matrix.
• Compute the dot product of the row and column, placing the resultant sum into the corresponding position in the product matrix.
• Repeat this process for every row-column pair until the entire product matrix is filled.

In our example, to find \( \mathbf{A} \cdot \mathbf{B} \), we calculate each element like \((\mathbf{A} \cdot \mathbf{B})_{11} = (3 \cdot 3) + (2 \cdot (-1)) = 7\). Each calculation follows this pattern: multiply, add results, and insert into the final matrix. This step-by-step methodology ensures accuracy in determining the matrix multiplication result.