Problem 34 - 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. \(C(A+B)=A C+B C\)

Step-by-Step Solution

Answer

The equation is false; \(C(A+B) \neq AC+BC\).

Step 1: Calculate A + B

First, we need to add matrices A and B. The matrix addition is performed element-wise. \[A + B = \begin{bmatrix} 1 + (-5) & -2 + 2 \ 4 + 4 & 3 + 3 \end{bmatrix} = \begin{bmatrix} -4 & 0 \ 8 & 6 \end{bmatrix}\]

Step 2: Calculate C(A + B)

Now, we multiply matrix C with the matrix we found, which is \(A + B\). Matrix multiplication involves taking the dot product of rows and columns.\[C(A+B) = \begin{bmatrix} 5 & 1 \ 2 & -4 \end{bmatrix} \begin{bmatrix} -4 & 0 \ 8 & 6 \end{bmatrix} = \begin{bmatrix} (5)(-4) + (1)(8) & (5)(0) + (1)(6) \ (2)(-4) + (-4)(8) & (2)(0) + (-4)(6) \end{bmatrix} = \begin{bmatrix} -12 & 6 \ -36 & -24 \end{bmatrix}\]

Step 3: Calculate AC and BC separately

To verify \(C(A+B)=AC+BC\), we need both \(AC\) and \(BC\). Start with \(AC\):\[AC = \begin{bmatrix} 1 & -2 \ 4 & 3 \end{bmatrix} \begin{bmatrix} 5 & 1 \ 2 & -4 \end{bmatrix} = \begin{bmatrix} (1)(5) + (-2)(2) & (1)(1) + (-2)(-4) \ (4)(5) + (3)(2) & (4)(1) + (3)(-4) \end{bmatrix} = \begin{bmatrix} 1 & 9 \ 26 & -8 \end{bmatrix}\]Next, calculate \(BC\):\[BC = \begin{bmatrix} -5 & 2 \ 4 & 3 \end{bmatrix} \begin{bmatrix} 5 & 1 \ 2 & -4 \end{bmatrix} = \begin{bmatrix} (-5)(5) + (2)(2) & (-5)(1) + (2)(-4) \ (4)(5) + (3)(2) & (4)(1) + (3)(-4) \end{bmatrix} = \begin{bmatrix} -21 & -13 \ 26 & -8 \end{bmatrix}\]

Step 4: Calculate AC + BC

Add the matrices \(AC\) and \(BC\):\[AC + BC = \begin{bmatrix} 1 & 9 \ 26 & -8 \end{bmatrix} + \begin{bmatrix} -21 & -13 \ 26 & -8 \end{bmatrix} = \begin{bmatrix} 1 + (-21) & 9 + (-13) \ 26 + 26 & -8 + (-8) \end{bmatrix} = \begin{bmatrix} -20 & -4 \ 52 & -16 \end{bmatrix}\]

Step 5: Compare C(A+B) with AC + BC

Compare the results of \(C(A+B)\) from Step 2 and \(AC+BC\) from Step 4:\[C(A+B) = \begin{bmatrix} -12 & 6 \ -36 & -24 \end{bmatrix}\]\[AC+BC = \begin{bmatrix} -20 & -4 \ 52 & -16 \end{bmatrix}\]Since these matrices are not equal, the equation \(C(A+B) = AC+BC\) is false.

Key Concepts

Matrix Addition

Matrix addition is the process of adding two matrices by adding their corresponding elements. This operation requires that the matrices involved have the same dimensions. To perform matrix addition:

• Ensure that both matrices have the same number of rows and columns.
• Add each element in one matrix to its corresponding element in the other matrix.

For example, if you have two matrices, say, \[ A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \] and \[ B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}, \] their sum, \( A + B \), would be calculated as:\[ A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}. \]The result is a new matrix where each element is the sum of the corresponding elements from the original matrices. This fundamental operation is crucial in solving linear equations, representing linear transformations, and facilitating other matrix operations like multiplication.

Matrix Scalar Multiplication

Matrix scalar multiplication involves multiplying every element of a matrix by a scalar (a constant number). This operation is simple yet powerful and useful in scaling transformations in mathematics and applied fields.To perform scalar multiplication:

• Take a matrix, \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \).
• Multiply each element of the matrix by the scalar, \( c \).

The result is:\[ cA = \begin{bmatrix} c\cdot a_{11} & c\cdot a_{12} \ c\cdot a_{21} & c\cdot a_{22} \end{bmatrix}. \]This method changes the size but not the shape of the matrix lattice in terms of data scaling. It is widely used when adjusting proportions in data analysis or transforming matrices in graphics.

Matrix Equality

Matrix equality is a condition where two matrices are considered equal if they have the same dimensions and identical corresponding entries. Understanding this concept is crucial when verifying if operations like matrix addition and multiplication result in equal matrices.Here’s how you can determine if two matrices are equal:

• Check if both matrices have the same number of rows and columns.
• Compare all corresponding elements from the two matrices. They must be exactly the same for equality.

If you have two matrices \( A \) and \( B \), they are equal if:\[ A = B \quad \text{only if} \quad a_{ij} = b_{ij} \quad \text{for all i, j}. \]For example, consider:\[ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}. \]Since all corresponding elements are equivalent, \( A = B \). If any element differs, the matrices are not equal. Matrix equality is crucial in validating calculations and solutions.

Algebraic Proofs

Proving algebraic properties involving matrices requires a logical sequence of operations to establish given identities or properties. Algebraic proofs in matrices often focus on operations like addition, multiplication, and the application of specific laws (such as distributive or associative laws).To construct algebraic proofs, follow these steps:

• Start by clearly defining the given identities and what you are required to prove.
• Perform the necessary matrix operations, just as you would solve traditional algebraic proofs.
• Ensure each step follows logically and verifies each claim with actual matrix computations.

Consider the equation addressed in our exercise:\[ C(A + B) ?= AC + BC \]By calculating each side separately and comparing results, you form an algebraic proof. The initial computations showed the matrices are different, hence:\[ C(A+B) eq AC+BC \]This discrepancy demonstrates the initial identity does not hold for the given matrices, providing a clear, logically structured proof through computation and comparison. Understanding and constructing these proofs hone your analytical skills and deepen comprehension of matrix operations.