Problem 8 - College Algebra

Question

Use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined. \(A=\left[\begin{array}{ll}1 & 3 \\ 0 & 7\end{array}\right], B=\left[\begin{array}{rr}2 & 14 \\ 22 & 6\end{array}\right], C=\left[\begin{array}{rr}1 & 5 \\ 8 & 92 \\ 12 & 6\end{array}\right], D=\left[\begin{array}{rr}10 & 14 \\ 7 & 2 \\ 5 & 61\end{array}\right], E=\left[\begin{array}{rr}6 & 12 \\ 14 & 5\end{array}\right], F=\left[\begin{array}{rr}0 & 9 \\ 78 & 17 \\ 15 & 4\end{array}\right]\) \(A+C\)

Step-by-Step Solution

Answer

The matrix addition \(A + C\) is undefined.

Step 1: Determine Matrix Dimensions

First, determine the dimensions of matrices \(A\) and \(C\). Matrix \(A\) is a \(2 \times 2\) matrix, and matrix \(C\) is a \(3 \times 2\) matrix.

Step 2: Check Compatibility for Addition

Matrix addition requires that both matrices have the same dimensions. Since \(A\) is a \(2 \times 2\) matrix and \(C\) is a \(3 \times 2\) matrix, they do not have the same dimensions. Therefore, the addition \(A + C\) is undefined.

Key Concepts

Matrix Addition

Matrix addition is a process where two matrices are combined to give a new matrix by adding the corresponding elements. For this process to be possible, both matrices must have the same dimensions. This means that if you have a matrix with dimensions of 2 rows and 2 columns, the other matrix you intend to add must also be 2 by 2.
Unlike simple number addition, each element in one matrix is added to its corresponding element in the other matrix. For instance, if you have a simple matrix, say:

• Matrix A: \[\begin{bmatrix} 1 & 3 \ 0 & 7 \end{bmatrix}\]
and another Matrix E:
• \[\begin{bmatrix} 6 & 12 \ 14 & 5 \end{bmatrix} \]
The result of adding these two matrices will be:
• \[\begin{bmatrix} 7 & 15 \ 14 & 12 \end{bmatrix} \]

Every spot in one matrix corresponds to the identical spot in the other matrix. If the matrices do not share the same dimensions, like in our exercise with matrices A and C, it is simply impossible to add them.

Matrix Subtraction

Matrix subtraction works similarly to matrix addition but instead involves taking away elements from corresponding positions. Just like in matrix addition, subtraction requires the matrices to be the same size in terms of rows and columns.
Imagine two matrices:

• \[\begin{bmatrix} 4 & 12 \ 14 & 3 \end{bmatrix} \]

and

• \[\begin{bmatrix} 2 & 4 \ 8 & 2 \end{bmatrix} \]

The result will be:

• \[\begin{bmatrix} 2 & 8 \ 6 & 1 \end{bmatrix} \]

Again, each element is subtracted from the corresponding position in another matrix. If an attempt is made to subtract matrices with different dimensions, like from our exercise when investigating A and C, the operation is undefined as well.

Matrix Dimensions

Understanding matrix dimensions is crucial for any kind of matrix operation. Dimensions refer to the number of rows and columns in a matrix. In mathematical notation, the dimensions of a matrix are represented as rows × columns, for example, a 2×3 matrix means it has 2 rows and 3 columns.
Matrix A from the exercise is a 2×2, which means:

• Two rows
• Two columns

Matrix C, on the other hand, is a 3×2:

• Three rows
• Two columns

The number of elements in a row doesn't necessarily match the number of elements in a column for these matrices. However, understanding this structure helps in figuring out which operations can be performed, ensuring you are on the right path to solving matrix equations effectively.

Compatibility of Matrices

Matrix compatibility is a term used to describe whether two matrices can be combined using particular operations such as addition or subtraction. The primary rule for these operations is that both matrices need to have identical dimensions.
For instance, two matrices that share structure 2×3 can be added or subtracted with ease. This structural compatibility ensures each element has another element in the same position to operate with.

• Equal number of rows
• Equal number of columns

In our exercise, matrices A and C are not compatible for addition or subtraction because A is a 2×2 matrix while C is a 3×2 matrix. Therefore, they lack the essential quality of having matching dimensions, making it impossible to perform these operations on them. Compatibility is an essential consideration before proceeding to attempt any matrix operations.