Problem 13 - Algebra 2

Question

Use \(A=\left[\begin{array}{cc}{2} & {-1} \\ {3} & {5}\end{array}\right], B=\left[\begin{array}{cc}{-4} & {1} \\ {8} & {0}\end{array}\right]\) and \(C=\left[\begin{array}{cc}{3} & {2} \\ {-1} & {2}\end{array}\right]\) to determine whether the following equations are true for the given matrices. \(A(B C)=(A B) C\)

Step-by-Step Solution

Answer

The equations are true; \( A(BC) = (AB)C \).

Step 1: Compute BC

First, calculate the product of matrices B and C.Matrix B:\[ B = \begin{bmatrix} -4 & 1 \ 8 & 0 \end{bmatrix} \]Matrix C:\[ C = \begin{bmatrix} 3 & 2 \ -1 & 2 \end{bmatrix} \]Perform the matrix multiplication \( BC \):\[ BC = \begin{bmatrix} (-4)(3) + (1)(-1) & (-4)(2) + (1)(2) \ (8)(3) + (0)(-1) & (8)(2) + (0)(2) \end{bmatrix} \]\[ BC = \begin{bmatrix} -12 - 1 & -8 + 2 \ 24 & 16 \end{bmatrix} \]\[ BC = \begin{bmatrix} -13 & -6 \ 24 & 16 \end{bmatrix} \]

Step 2: Compute A(BC)

Next, find the product of matrix A and the result of \( BC \) obtained in Step 1.Matrix A:\[ A = \begin{bmatrix} 2 & -1 \ 3 & 5 \end{bmatrix} \]The resulting matrix from step 1:\[ BC = \begin{bmatrix} -13 & -6 \ 24 & 16 \end{bmatrix} \]Calculate \( A(BC) \):\[ A(BC) = \begin{bmatrix} (2)(-13) + (-1)(24) & (2)(-6) + (-1)(16) \ (3)(-13) + (5)(24) & (3)(-6) + (5)(16) \end{bmatrix} \]\[ A(BC) = \begin{bmatrix} -26 - 24 & -12 - 16 \ -39 + 120 & -18 + 80 \end{bmatrix} \]\[ A(BC) = \begin{bmatrix} -50 & -28 \ 81 & 62 \end{bmatrix} \]

Step 3: Compute AB

Now, compute the product of matrices A and B.Matrix A:\[ A = \begin{bmatrix} 2 & -1 \ 3 & 5 \end{bmatrix} \]Matrix B:\[ B = \begin{bmatrix} -4 & 1 \ 8 & 0 \end{bmatrix} \]Perform the multiplication \( AB \):\[ AB = \begin{bmatrix} (2)(-4) + (-1)(8) & (2)(1) + (-1)(0) \ (3)(-4) + (5)(8) & (3)(1) + (5)(0) \end{bmatrix} \]\[ AB = \begin{bmatrix} -8 - 8 & 2 \ (12) & (3) \end{bmatrix} \]\[ AB = \begin{bmatrix} -16 & 2 \ 28 & 3 \end{bmatrix} \]

Step 4: Compute (AB)C

Finally, find the product of the result from Step 3 and matrix C.The resulting matrix from step 3:\[ AB = \begin{bmatrix} -16 & 2 \ 28 & 3 \end{bmatrix} \]Matrix C:\[ C = \begin{bmatrix} 3 & 2 \ -1 & 2 \end{bmatrix} \]Calculate \( (AB)C \):\[ (AB)C = \begin{bmatrix} (-16)(3) + (2)(-1) & (-16)(2) + (2)(2) \ (28)(3) + (3)(-1) & (28)(2) + (3)(2) \end{bmatrix} \]\[ (AB)C = \begin{bmatrix} -48 - 2 & -32 + 4 \ 84 - 3 & 56 + 6 \end{bmatrix} \]\[ (AB)C = \begin{bmatrix} -50 & -28 \ 81 & 62 \end{bmatrix} \]

Step 5: Conclusion: Verify Equality

Now that both sides of the equation have been calculated, we can compare them. From Steps 2 and 4:\( A(BC) = \begin{bmatrix} -50 & -28 \ 81 & 62 \end{bmatrix} \) \( (AB)C = \begin{bmatrix} -50 & -28 \ 81 & 62 \end{bmatrix} \)Since these matrices are equal, the equation \( A(BC) = (AB)C \) holds true for the given matrices.

Key Concepts

Associative Property of Matrices

The associative property is fundamental to understanding matrix multiplication.
It tells us that when we multiply three matrices together, the product is the same no matter how we group them.
For example, when you have matrices \( A \), \( B \), and \( C \), the associative property ensures that:

• \( (AB)C = A(BC) \)

This allows us to perform matrix multiplication in different sequences without affecting the final result.

In matrix multiplication, unlike addition, specific order is crucial, but thanks to the associative property, we are free to decide which pair of matrices to multiply first.
This is especially useful in computing environments where optimization of multiplication sequence can lead to improved performance.

To sum up, understanding and applying the associative property in matrices helps streamline calculations and ensures consistency in results.
This property is one of the building blocks in the broader field of matrix algebra.

Matrix Equality

Matrix equality serves as the basis for verifying calculations in matrix algebra.
Two matrices are considered equal if they have the same dimensions and identical elements in corresponding positions.

• Each element in the first row of Matrix \( A \) must exactly match the corresponding element in the first row of Matrix \( B \), and so on for all rows and columns.

In practical terms, when you're solving problems involving matrix multiplication, as demonstrated in the exercise, confirming equality is essential.
After calculating both \( A(BC) \) and \( (AB)C \), checking that the two resulting matrices are identical helps validate your use of the associative property.

Achieving matrix equality also means ensuring accuracy during computations.
Each step of multiplication must be performed with care to maintain the values of elements from start to finish.
By comparing matrices for equality, we assure that mathematical principles are consistently applied, reinforcing the correct relationships between matrices.

Matrix Algebra

Matrix algebra is an area of mathematics where matrices are studied and manipulated using algebraic operations.
It extends the familiar algebraic principles you may be used to, involving numbers but instead applies them to matrices.
Some fundamental operations in matrix algebra include:

• Matrix Addition and Subtraction: These operations are straightforward and involve adding or subtracting corresponding elements from matrices of the same dimensions.
• Matrix Multiplication: As showcased in our exercise, matrix multiplication is more complex and involves summing the products of row elements with column elements.
• Scalar Multiplication: Here, every element of a matrix is multiplied by a scalar (a single number).

Unlike regular algebra, matrix algebra has unique properties such as non-commutativity in multiplication. This means \( AB \) does not necessarily equal \( BA \).
Understanding these distinct traits is critical in exploring more advanced topics in mathematics and computer science.
Matrix algebra plays a key role in various fields including physics, engineering, and computer graphics.
Mastery of matrix operations and their properties opens the door to solving complex problems involving systems of equations and transformations.