Problem 31 - 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. \(c(A B)=A(c B)\)

Step-by-Step Solution

Answer

The equation \( c(AB) = A(cB) \) is true.

Step 1: Calculate c(AB)

First, calculate the product of matrices A and B, and then multiply the resulting matrix by the scalar c.Matrix A: \[ A = \begin{bmatrix} 1 & -2 \ 4 & 3 \end{bmatrix} \]Matrix B: \[ B = \begin{bmatrix} -5 & 2 \ 4 & 3 \end{bmatrix} \]The product AB is calculated by multiplying each element using the dot product as follows:\[ AB = \begin{bmatrix} (1)(-5) + (-2)(4) & (1)(2) + (-2)(3) \ (4)(-5) + (3)(4) & (4)(2) + (3)(3) \end{bmatrix} = \begin{bmatrix} -5 - 8 & 2 - 6 \ -20 + 12 & 8 + 9 \end{bmatrix} = \begin{bmatrix} -13 & -4 \ -8 & 17 \end{bmatrix} \]Multiply by scalar c:\[ c(AB) = 3 \begin{bmatrix} -13 & -4 \ -8 & 17 \end{bmatrix} = \begin{bmatrix} 3(-13) & 3(-4) \ 3(-8) & 3(17) \end{bmatrix} = \begin{bmatrix} -39 & -12 \ -24 & 51 \end{bmatrix} \]

Step 2: Calculate A(cB)

Multiply matrix B by the scalar c, and then multiply the resulting matrix by matrix A.Scalar multiplication of B:\[ cB = 3 \begin{bmatrix} -5 & 2 \ 4 & 3 \end{bmatrix} = \begin{bmatrix} 3(-5) & 3(2) \ 3(4) & 3(3) \end{bmatrix} = \begin{bmatrix} -15 & 6 \ 12 & 9 \end{bmatrix} \]Now calculate the product A(cB):\[ A(cB) = \begin{bmatrix} 1 & -2 \ 4 & 3 \end{bmatrix} \begin{bmatrix} -15 & 6 \ 12 & 9 \end{bmatrix} = \begin{bmatrix} (1)(-15) + (-2)(12) & (1)(6) + (-2)(9) \ (4)(-15) + (3)(12) & (4)(6) + (3)(9) \end{bmatrix} \]\[ = \begin{bmatrix} -15 - 24 & 6 - 18 \ -60 + 36 & 24 + 27 \end{bmatrix} = \begin{bmatrix} -39 & -12 \ -24 & 51 \end{bmatrix} \]

Step 3: Compare Results

Compare the matrices obtained from the two methods to determine if they are equal.From Step 1, we have:\[ c(AB) = \begin{bmatrix} -39 & -12 \ -24 & 51 \end{bmatrix} \]From Step 2, we have:\[ A(cB) = \begin{bmatrix} -39 & -12 \ -24 & 51 \end{bmatrix} \]Both matrices are identical.

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in matrix algebra. It involves multiplying each element of a matrix by a scalar, which is simply a real number. This process stretches or shrinks the matrix but does not change its overall structure.
For example, consider a scalar \( c = 3 \) and a matrix \( B = \begin{bmatrix} -5 & 2 \ 4 & 3 \end{bmatrix} \). In scalar multiplication, we compute \( cB = 3 \times \begin{bmatrix} -5 & 2 \ 4 & 3 \end{bmatrix} = \begin{bmatrix} 3(-5) & 3(2) \ 3(4) & 3(3) \end{bmatrix} = \begin{bmatrix} -15 & 6 \ 12 & 9 \end{bmatrix} \).
The result is a matrix where each element of the original matrix is multiplied by the scalar. This operation maintains the shape of the original matrix; thus, the number of rows and columns remains unchanged.

Matrix Algebra

Matrix algebra is a collection of mathematical techniques and operations for dealing with matrices. It involves various operations like addition, subtraction, multiplication, and inversion of matrices.
In matrix multiplication, two matrices can be multiplied if the number of columns in the first matrix equals the number of rows in the second matrix. For example, when multiplying \( A = \begin{bmatrix} 1 & -2 \ 4 & 3 \end{bmatrix} \) with \( B = \begin{bmatrix} -5 & 2 \ 4 & 3 \end{bmatrix} \), each element in the resulting matrix \( AB \) is derived by performing the dot product of the rows from \( A \) with the columns of \( B \).
Matrix algebra principles also include distributing scalar multiplication over matrix multiplication. The equation \( c(AB) = A(cB) \) illustrates this property. Here, we see that whether we multiply the product \( AB \) by \( c \) or multiply \( B \) by \( c \) first and then perform the matrix multiplication with \( A \), the results are equivalent.

Matrix Equivalence

Matrix equivalence refers to situations where matrices are structurally and numerically identical. Two matrices are considered equivalent if they have the same dimensions and their corresponding elements are equal.
In the given exercise, after performing operations \( c(AB) \) and \( A(cB) \), we compare the resulting matrices. When both outcomes are \( \begin{bmatrix} -39 & -12 \ -24 & 51 \end{bmatrix} \), we confirm that these matrices are equivalent because their corresponding elements match perfectly. This equivalence check is crucial in verifying the correctness of operations and shows consistent results across different calculation methods.
Thus, matrix equivalence ensures that mathematical properties like associative, distributive, and commutative laws in matrix algebra hold true, providing a strong foundation for more complex operations.