Problem 26 - College Algebra with Corequisite Support

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}2 & -5 \\ 6 & 7\end{array}\right], B=\left[\begin{array}{ll}-9 & 6 \\ -4 & 2\end{array}\right], C=\left[\begin{array}{ll}0 & 9 \\ 7 & 1\end{array}\right], D=\left[\begin{array}{rrr}-8 & 7 & -5 \\ 4 & 3 & 2 \\ 0 & 9 & 2\end{array}\right], E=\left[\begin{array}{rrr}4 & 5 & 3 \\ 7 & -6 & -5 \\ 1 & 0 & 9\end{array}\right]\) \(2 C+B\)

Step-by-Step Solution

Answer

The result is \(\begin{bmatrix} -9 & 24 \\ 10 & 4 \end{bmatrix}\).

Step 1: Verify Dimensions for Addition

To perform the addition of matrices, they must have the same dimensions. Matrix \(C\) is a \(2 \times 2\) matrix and matrix \(B\) is also a \(2 \times 2\) matrix. Thus, they can be added together.

Step 2: Calculate Scalar Multiplication \(2C\)

Multiply each element of matrix \(C\) by 2: \[2C = 2 \times \begin{bmatrix} 0 & 9 \ 7 & 1 \end{bmatrix} = \begin{bmatrix} 0 \times 2 & 9 \times 2 \ 7 \times 2 & 1 \times 2 \end{bmatrix} = \begin{bmatrix} 0 & 18 \ 14 & 2 \end{bmatrix}\]

Step 3: Add Matrices \(2C\) and \(B\)

Now add the matrices \(2C\) and \(B\):\[\begin{bmatrix} 0 & 18 \ 14 & 2 \end{bmatrix} + \begin{bmatrix} -9 & 6 \ -4 & 2 \end{bmatrix} = \begin{bmatrix} 0 + (-9) & 18 + 6 \ 14 + (-4) & 2 + 2 \end{bmatrix} = \begin{bmatrix} -9 & 24 \ 10 & 4 \end{bmatrix}\]

Step 4: Conclusion

The resulting matrix after performing the operations \(2C + B\) is \(\begin{bmatrix} -9 & 24 \ 10 & 4 \end{bmatrix}\).

Key Concepts

Matrix Addition

Matrix addition is a straightforward process where corresponding elements of two matrices are added together. For matrices to be added, they must share the same dimensions. This means if one matrix is a 2x2 matrix, the other must also be a 2x2 matrix.
To visualize this, think of each element in a matrix as a position in a grid. You can only add elements that lie in the same position on both matrices.

• If Matrix A = \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) and Matrix B = \(\begin{bmatrix} e & f \ g & h \end{bmatrix}\), the resulting matrix C after addition will be \(\begin{bmatrix} a+e & b+f \ c+g & d+h \end{bmatrix}\).
• Remember, the order of matrices matters as they need to perfectly overlap.

Ensuring the matrices have the right dimensions is the first step whenever you're asked to add two matrices. If they don't match, the operation is not possible.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a scalar, which is a single number. This operation is like scaling the entire matrix up or down, depending on whether the scalar is greater or less than one.
For instance, if you multiply matrix \(\begin{bmatrix} 0 & 9 \ 7 & 1 \end{bmatrix}\) by 2, you simply multiply each element individually:

• The first element becomes \(0 \times 2 = 0\).
• The second element becomes \(9 \times 2 = 18\).
• The third element becomes \(7 \times 2 = 14\).
• The fourth element becomes \(1 \times 2 = 2\).

After multiplying, every value in the matrix is twice as large as it initially was. Scalar multiplication helps scale entire datasets or modify individual matrix properties by altering each part by the same factor.

Matrix Dimensions

Understanding matrix dimensions is crucial for performing various matrix operations, such as addition or multiplication. The dimensions of a matrix are given as \(m \times n\), where \(m\) represents the number of rows and \(n\) represents the number of columns.
This notation helps you ascertain how data is structured within the matrix. For example, a 2x2 matrix like \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) has two rows and two columns.

• In general, the dimensions tell you how many entries fit in each direction.
• The dimensions must match in matrix addition because each element in one matrix needs a corresponding element in the second matrix to be added.

When performing operations, always double-check dimensions first to know if the operation is possible and to predict the potential outcome's dimensions. This is fundamental to navigating more complex matrix operations as you advance.