Question

\\[ \text { Given } A=\left[\begin{array}{ll} 2 & 8 \\ 3 & 0 \\ 5 & 1 \end{array}\right], B=\left[\begin{array}{ll} 2 & 0 \\ 3 & 8 \end{array}\right], \text { and } C=\left[\begin{array}{ll} 7 & 2 \\ 6 & 3 \end{array}\right]: \\] (a) Is \(A B\) defined? Calculate \(A B\). Can you calculate \(8 A\) ? Why? (b) Is \(B C\) defined? Calculate \(B C\). Is CB defined? If so, calculate \(C B\). Is it true that \(B C=C B ?\)

Short answer

(a) AB is defined and \( AB = \left[ \begin{array}{cc} 28 & 64 \\ 6 & 0 \\ 13 & 8 \end{array} \right] \); 8A is \( \left[ \begin{array}{cc} 16 & 64 \\ 24 & 0 \\ 40 & 8 \end{array} \right] \). (b) BC is defined with \( BC = \left[ \begin{array}{cc} 14 & 4 \\ 69 & 30 \end{array} \right] \), and CB is defined with \( CB = \left[ \begin{array}{cc} 20 & 16 \\ 21 & 24 \end{array} \right] \); BC ≠ CB.

Step-by-step solution

Determine if AB is defined

For the product of two matrices to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix \( A \) is a \(3 \times 2\) matrix and matrix \( B \) is a \(2 \times 2\) matrix. Since the number of columns in \( A \) (2) equals the number of rows in \( B \) (2), the product \( AB \) is defined.

Calculate Matrix Product AB

To find \( AB \), multiply each row of \( A \) with each column of \( B \):\[ AB = \left[ \begin{array}{cc} 2 \cdot 2 + 8 \cdot 3 & 2 \cdot 0 + 8 \cdot 8 \ 3 \cdot 2 + 0 \cdot 3 & 3 \cdot 0 + 0 \cdot 8 \ 5 \cdot 2 + 1 \cdot 3 & 5 \cdot 0 + 1 \cdot 8 \end{array} \right] = \left[ \begin{array}{cc} 28 & 64 \ 6 & 0 \ 13 & 8 \end{array} \right] \]

Check if 8A is possible and calculate

Scalar multiplication is always defined for matrices. To calculate \( 8A \), multiply each element of \( A \) by 8:\[ 8A = 8 \times \left[ \begin{array}{cc} 2 & 8 \ 3 & 0 \ 5 & 1 \end{array} \right] = \left[ \begin{array}{cc} 16 & 64 \ 24 & 0 \ 40 & 8 \end{array} \right] \]

Determine if BC is defined

Matrix \( B \) is a \(2 \times 2\) matrix and matrix \( C \) is a \(2 \times 2\) matrix. The number of columns in \( B \) (2) equals the number of rows in \( C \) (2), so the product \( BC \) is defined.

Calculate Matrix Product BC

To compute \( BC \), multiply each row of \( B \) by each column of \( C \):\[ BC = \left[ \begin{array}{cc} 2 \cdot 7 + 0 \cdot 6 & 2 \cdot 2 + 0 \cdot 3 \ 3 \cdot 7 + 8 \cdot 6 & 3 \cdot 2 + 8 \cdot 3 \end{array} \right] = \left[ \begin{array}{cc} 14 & 4 \ 69 & 30 \end{array} \right] \]

Determine if CB is defined

Matrix \( C \) is a \(2 \times 2\) matrix and matrix \( B \) is a \(2 \times 2\) matrix. The number of columns in \( C \) (2) matches the number of rows in \( B \) (2), so the product \( CB \) is defined.

Calculate Matrix Product CB

To find \( CB \), multiply each row of \( C \) by each column in \( B \):\[ CB = \left[ \begin{array}{cc} 7 \cdot 2 + 2 \cdot 3 & 7 \cdot 0 + 2 \cdot 8 \ 6 \cdot 2 + 3 \cdot 3 & 6 \cdot 0 + 3 \cdot 8 \end{array} \right] = \left[ \begin{array}{cc} 20 & 16 \ 21 & 24 \end{array} \right] \]

Compare BC and CB

The matrices \( BC = \left[ \begin{array}{cc} 14 & 4 \ 69 & 30 \end{array} \right] \) and \( CB = \left[ \begin{array}{cc} 20 & 16 \ 21 & 24 \end{array} \right] \) are different, proving that in this case, \( BC eq CB \).

Key concepts

Matrix Dimensions

Understanding the concept of matrix dimensions is crucial for performing any type of matrix operations. A matrix's dimensions refer to its size, expressed as \( m \times n \), where \( m \) is the number of rows, and \( n \) is the number of columns. For instance, matrix \( A \) in the provided exercise is a \( 3 \times 2 \) matrix because it has three rows and two columns.
It's essential to grasp these dimensions when considering whether matrices can be multiplied. The pivotal rule is that the number of columns in the first matrix must equal the number of rows in the second matrix for the multiplication to be defined. This rule is why the product \( AB \) is possible, with \( A \) being \( 3 \times 2 \) and \( B \) being \( 2 \times 2 \), since the two columns of \( A \) correspond to the two rows of \( B \).
However, when considering different matrices' multiplication, such as \( BC \) or \( CB \), matching dimensions becomes your primary check before carrying out multiplication. So, always take note of a matrix's size to prevent errors in multiplication.

Scalar Multiplication

Scalar multiplication is one of the more straightforward operations involving matrices. Unlike matrix multiplication, it involves multiplying every element of a matrix by a single number, known as a scalar. This operation is always defined since it does not depend on the dimensions of another matrix.
In the exercise you reviewed, multiplying the matrix \( A \) by the scalar 8 means each element in \( A \) is scaled by 8. So, if a specific element in \( A \) is 2, it becomes \( 2 imes 8 = 16 \) in the new matrix, \( 8A \).
Scalar multiplication is especially useful when you want to represent a matrix scaled up or down in size without changing its essential properties. It’s akin to zooming into or out of an image, making every detail proportionately larger or smaller.

Matrix Product

The matrix product, or matrix multiplication, occurs when you multiply two matrices. It's a bit more complex than simple arithmetic. For the matrix product to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
The actual multiplication process involves taking each row of the first matrix and multiplying it by each column of the second. For each pair of row and column, you find the sum of the products of their corresponding elements. This multiplication results in a new matrix. In your exercise, you've seen the multiplication of \( AB \) and \( BC \) as examples of this process.
Matrix products are not commutative, meaning \( AB \) is not necessarily equal to \( BA \). Each product varies based on the original matrices' arrangement, underlining the importance of the order of multiplication.

Associative Property of Matrices

The associative property is a key attribute in matrix multiplication, but it often confuses students. The associative property means that the way you group matrices during multiplication doesn't affect the product. In simpler terms, if you have three matrices \( A \), \( B \), and \( C \), then \( (AB)C = A(BC) \).
However, despite the associative property, not every pair of matrices can be multiplied in any order, as discussed with the matrix product differences. You must first ensure that the matrices in each individual multiplication are compatible with each other's dimensions.
In the exercise, though \( BC \) and \( CB \) are both defined, they are not equal, demonstrating a different property: matrix multiplication is not commutative. But, with three compatible matrices, associative grouping gives flexibility in how you perform multiplications without changing results, provided dimensions are respected for each step.