Problem 33 - Algebra for College Students

Question

For Problems \(31-34\), use the following matrices. \(\begin{aligned} &A=\left[\begin{array}{rr} 2 & 4 \\ 5 & -3 \end{array}\right] \quad B=\left[\begin{array}{ll} -2 & 3 \\ -1 & 2 \end{array}\right] \\ &C=\left[\begin{array}{ll} 2 & 1 \\ 3 & 7 \end{array}\right] \end{aligned}\) $$ \text { Show that }(A+B) C=A C+B C \text {. } $$

Step-by-Step Solution

Answer

This verifies that \((A + B) C = A C + B C\) as both sides are equal.

Step 1: Calculate A + B

First, we find the sum of matrices \(A\) and \(B\).\[A + B = \begin{bmatrix} 2 & 4 \ 5 & -3 \end{bmatrix} + \begin{bmatrix} -2 & 3 \ -1 & 2 \end{bmatrix} = \begin{bmatrix} 2 + (-2) & 4 + 3 \ 5 + (-1) & -3 + 2 \end{bmatrix} = \begin{bmatrix} 0 & 7 \ 4 & -1 \end{bmatrix} \]

Step 2: Calculate (A + B)C

Now, multiply the result of \( A + B \) by matrix \( C \).\[(A + B)C = \begin{bmatrix} 0 & 7 \ 4 & -1 \end{bmatrix} \begin{bmatrix} 2 & 1 \ 3 & 7 \end{bmatrix} = \begin{bmatrix} (0\cdot2 + 7\cdot3) & (0\cdot1 + 7\cdot7) \ (4\cdot2 + (-1)\cdot3) & (4\cdot1 + (-1)\cdot7) \end{bmatrix} = \begin{bmatrix} 21 & 49 \ 5 & -3 \end{bmatrix} \]

Step 3: Calculate AC

Multiply matrix \(A\) by matrix \(C\).\[AC = \begin{bmatrix} 2 & 4 \ 5 & -3 \end{bmatrix} \begin{bmatrix} 2 & 1 \ 3 & 7 \end{bmatrix} = \begin{bmatrix} (2\cdot2 + 4\cdot3) & (2\cdot1 + 4\cdot7) \ (5\cdot2 + (-3)\cdot3) & (5\cdot1 + (-3)\cdot7) \end{bmatrix} = \begin{bmatrix} 16 & 30 \ 1 & -16 \end{bmatrix} \]

Step 4: Calculate BC

Multiply matrix \(B\) by matrix \(C\).\[BC = \begin{bmatrix} -2 & 3 \ -1 & 2 \end{bmatrix} \begin{bmatrix} 2 & 1 \ 3 & 7 \end{bmatrix} = \begin{bmatrix} ((-2)\cdot2 + 3\cdot3) & ((-2)\cdot1 + 3\cdot7) \ ((-1)\cdot2 + 2\cdot3) & ((-1)\cdot1 + 2\cdot7) \end{bmatrix} = \begin{bmatrix} 5 & 19 \ 4 & 13 \end{bmatrix} \]

Step 5: Calculate AC + BC

Add matrices \(AC\) and \(BC\) together to verify the equation.\[AC + BC = \begin{bmatrix} 16 & 30 \ 1 & -16 \end{bmatrix} + \begin{bmatrix} 5 & 19 \ 4 & 13 \end{bmatrix} = \begin{bmatrix} 16 + 5 & 30 + 19 \ 1 + 4 & -16 + 13 \end{bmatrix} = \begin{bmatrix} 21 & 49 \ 5 & -3 \end{bmatrix} \]

Step 6: Conclusion

Since \( (A + B)C = AC + BC \) results in the same matrix \( \begin{bmatrix} 21 & 49 \ 5 & -3 \end{bmatrix} \), the given equation is verified.

Key Concepts

Matrix Addition

Matrix addition is a key operation in matrix algebra. It involves adding corresponding elements of two matrices to form a new matrix. For two matrices to be added, they must have the same dimensions, meaning they must have the same number of rows and columns. In this exercise, we added matrices \(A\) and \(B\), both of which are \(2 \times 2\) matrices. The result was a new matrix: \[(A + B) = \begin{bmatrix} 0 & 7 \ 4 & -1 \end{bmatrix} \]. This result is obtained by adding each element of matrix \(A\) to the corresponding element of matrix \(B\):

• \(2 + (-2) = 0\)
• \(4 + 3 = 7\)
• \(5 + (-1) = 4\)
• \(-3 + 2 = -1\)

This principle of element-wise addition ensures that only matrices of the same size can be added together. It is a foundational concept that leads to more complex operations involving matrices.

Matrix Multiplication

Matrix multiplication is more complex than matrix addition and involves multiplying rows by columns. For two matrices \(A\) and \(B\), matrix multiplication \(AB\) is only possible if the number of columns in \(A\) is equal to the number of rows in \(B\). In our exercise, we calculated \((A + B)C\), \(AC\), and \(BC\), all involving multiplication with matrix \(C\), which is also a \(2 \times 2\) matrix. When performing matrix multiplication, each element in the resulting matrix is calculated by:

• Taking the dot product of the rows of the first matrix with the columns of the second matrix.
• For \((A + B)C\), for instance, we multiplied the first row of \((A + B)\) with each column of \(C\).

This gives us the expression: \[(A + B)C = \begin{bmatrix} 21 & 49 \ 5 & -3 \end{bmatrix} \]. Understanding the structure of matrix multiplication is key for advancing in matrix operations.

Distributive Property

The distributive property is central in verifying matrix equations like \((A + B)C = AC + BC\). This property states that for matrices \(A\), \(B\), and \(C\), the formula equals when both sides are evaluated independently. First, we calculated \((A + B)C\) using matrix multiplication. The resulting matrix \[\begin{bmatrix} 21 & 49 \ 5 & -3 \end{bmatrix}\] is obtained by multiplying the matrices as previously explained. Then, we computed \(AC\) and \(BC\) separately, followed by adding these results:

• \(AC\) equals \[\begin{bmatrix} 16 & 30 \ 1 & -16 \end{bmatrix}\]
• \(BC\) equals \[\begin{bmatrix} 5 & 19 \ 4 & 13 \end{bmatrix}\]

Adding them gives the same result as \((A + B)C\), thus confirming the distributive property: \[AC + BC = \begin{bmatrix} 21 & 49 \ 5 & -3 \end{bmatrix}\]. This property simplifies solving matrix equations.

Verification of Matrix Equations

Verification of matrix equations involves checking if an algebraic identity holds true by calculating both sides of the equation separately. In the exercise, the identity \((A + B)C = AC + BC\) was verified by performing explicit calculations:

• First, compute the left side, \((A + B)C\), using matrix addition and multiplication.
• Then, compute the right side by individually finding \(AC\) and \(BC\), and then adding the results.

The final step involves comparing both sides to see if they produce the same matrix. For our matrices, both expressions result in \[\begin{bmatrix} 21 & 49 \ 5 & -3 \end{bmatrix}\]. The confirmation demonstrates an understanding of matrix addition, multiplication, and the distributive property. Such verification processes ensure that matrix identities work universally, serving as a solid assurance for mathematical operations in matrix algebra.