Problem 33 - Algebra 2

Question

Use \(A=\left[\begin{array}{cc}{1} & {-2} \\ {4} & {3}\end{array}\right], B=\left[\begin{array}{cc}{-5} & {2} \\ {4} & {3}\end{array}\right], C=\left[\begin{array}{cc}{5} & {1} \\ {2} & {-4}\end{array}\right]\) and scalar \(c=3\) to determine whether the following equations are true for the given matrices. \(A C+B C=(A+B) C\)

Step-by-Step Solution

Answer

Equation is true; \(AC + BC = (A + B)C\).

Step 1: Calculate AC

Multiply matrix \(A\) by matrix \(C\). Use the matrix multiplication formula which involves summing the products of elements from the rows of the first matrix with columns of the second matrix. The result is:\[AC = \begin{bmatrix} 1 \times 5 + (-2) \times 2 & 1 \times 1 + (-2) \times (-4) \ 4 \times 5 + 3 \times 2 & 4 \times 1 + 3 \times (-4) \end{bmatrix} = \begin{bmatrix} 5 - 4 & 1 + 8 \ 20 + 6 & 4 - 12 \end{bmatrix} = \begin{bmatrix} 1 & 9 \ 26 & -8 \end{bmatrix}\]

Step 2: Calculate BC

Multiply matrix \(B\) by matrix \(C\). Again, apply the matrix multiplication rule.The result is:\[BC = \begin{bmatrix} (-5) \times 5 + 2 \times 2 & (-5) \times 1 + 2 \times (-4) \ 4 \times 5 + 3 \times 2 & 4 \times 1 + 3 \times (-4) \end{bmatrix} = \begin{bmatrix} -25 + 4 & -5 - 8 \ 20 + 6 & 4 - 12 \end{bmatrix} = \begin{bmatrix} -21 & -13 \ 26 & -8 \end{bmatrix}\]

Step 3: Calculate (AC + BC)

Add the results from Steps 1 and 2 to find \(AC + BC\).\[AC + BC = \begin{bmatrix} 1 & 9 \ 26 & -8 \end{bmatrix} + \begin{bmatrix} -21 & -13 \ 26 & -8 \end{bmatrix} = \begin{bmatrix} 1 - 21 & 9 - 13 \ 26 + 26 & -8 - 8 \end{bmatrix} = \begin{bmatrix} -20 & -4 \ 52 & -16 \end{bmatrix}\]

Step 4: Calculate A + B

Add matrix \(A\) and matrix \(B\).\[A + B = \begin{bmatrix} 1 & -2 \ 4 & 3 \end{bmatrix} + \begin{bmatrix} -5 & 2 \ 4 & 3 \end{bmatrix} = \begin{bmatrix} 1 - 5 & -2 + 2 \ 4 + 4 & 3 + 3 \end{bmatrix} = \begin{bmatrix} -4 & 0 \ 8 & 6 \end{bmatrix}\]

Step 5: Calculate (A + B)C

Multiply the result from Step 4 by matrix \(C\).\[(A + B)C = \begin{bmatrix} -4 & 0 \ 8 & 6 \end{bmatrix} \cdot \begin{bmatrix} 5 & 1 \ 2 & -4 \end{bmatrix} = \begin{bmatrix} -4 \times 5 + 0 \times 2 & -4 \times 1 + 0 \times (-4) \ 8 \times 5 + 6 \times 2 & 8 \times 1 + 6 \times (-4) \end{bmatrix} = \begin{bmatrix} -20 & -4 \ 40 + 12 & 8 - 24 \end{bmatrix} = \begin{bmatrix} -20 & -4 \ 52 & -16 \end{bmatrix}\]

Step 6: Conclusion: Compare (AC + BC) with (A + B)C

Check if the calculated \(AC + BC\) from Step 3 is equal to the calculated \((A + B)C\) from Step 5.Both results are:\[\begin{bmatrix} -20 & -4 \ 52 & -16 \end{bmatrix}\]Since they are equal, the equation is true.

Key Concepts

Matrix Addition

Matrix addition is a fundamental concept in linear algebra and typically the first operation students encounter when working with matrices. It involves adding two matrices to create a new matrix. In our exercise, the matrices have been given as:
- Matrix A = \( \begin{bmatrix} 1 & -2 \ 4 & 3 \end{bmatrix} \)- Matrix B = \( \begin{bmatrix} -5 & 2 \ 4 & 3 \end{bmatrix} \)
When adding matrices:

• Both matrices must have the same dimensions. In this case, both are 2x2 matrices.
• Add corresponding elements. For example, the top left element of \( A + B \) is \( 1 + (-5) = -4 \).

This operation is pretty straightforward, ensuring that we simply align the matrices and perform element-wise addition. The resulting matrix from adding A and B was given as \( \begin{bmatrix} -4 & 0 \ 8 & 6 \end{bmatrix} \). Matrix addition is crucial in various applications, like solving systems of equations and computer graphics.

Associative Property of Matrices

The associative property is a hallmark feature of various algebraic structures, including matrices. This property states that the grouping of matrices does not affect the result of their addition or multiplication. In mathematical terms, for any matrices \(A\), \(B\), and \(C\) that are suitably sized (i.e., have dimensions that align), this property implies:

• For addition: \((A + B) + C = A + (B + C)\)
• For multiplication: \((AB)C = A(BC)\)

In the given exercise, we focused on the multiplication aspect, demonstrating how \((A+B)C\) was compared with \(AC + BC\) to show they are equivalent. What’s important to note here is that due to this property, one can group operations in any order to make computation easier without affecting the final outcome. The associative property of multiplication was showcased effectively as both operations resulted in the same matrix: \( \begin{bmatrix} -20 & -4 \ 52 & -16 \end{bmatrix} \). It highlights the flexibility and consistency that associative operations offer in matrix computations.

Matrix Operations

Matrix operations are extensive and essential in understanding matrix algebra and its real-world applications. The primary matrix operations include addition, subtraction, multiplication, and scalar multiplication.

In our exercise:

• **Matrix Addition**: As discussed, it involved summing corresponding entries of the matrices A and B.
• **Matrix Multiplication**: This is quite different from addition. It's not merely an element-wise operation. Instead, the entry in the resulting matrix is the dot product of corresponding rows and columns from the two matrices being multiplied. For example, in \(AC\), to find the top-left element, we compute \(1 \cdot 5 + (-2) \cdot 2\).
• **Scalar Multiplication**: While not directly part of this problem, it involves multiplying every element of a matrix by a number, which is a scalar.

Understanding these operations provides a solid foundation for more complex topics in mathematics and helps solve practical problems in physics, engineering, and computer science. In the step-by-step exercise, you saw how mastering these operations can quickly verify matrix equations.