Question

Given \(A=\left[\begin{array}{ll}3 & 6 \\ 2 & 4\end{array}\right], B=\left[\begin{array}{rr}-1 & 7 \\ 8 & 4\end{array}\right],\) and \(C=\left[\begin{array}{ll}3 & 4 \\ 1 & 9\end{array}\right],\) verify that \((a)(A+B)+C=A+(B+C)\) (b) \((A+B)-C=A+(B-C)\)

Short answer

Both equations (a) and (b) are verified. Matrix operations confirm their validity.

Step-by-step solution

Add Matrices A and B

To find \( A + B \), sum the corresponding elements of matrices \( A \) and \( B \). Mathematically, this is done as follows: \[A + B = \begin{bmatrix} 3 + (-1) & 6 + 7 \ 2 + 8 & 4 + 4 \end{bmatrix} = \begin{bmatrix} 2 & 13 \ 10 & 8 \end{bmatrix}\]

Add Matrices A+B and C

Add matrix \( C \) to the result of \( A + B \) calculated in Step 1. \[(A + B) + C = \begin{bmatrix} 2 & 13 \ 10 & 8 \end{bmatrix} + \begin{bmatrix} 3 & 4 \ 1 & 9 \end{bmatrix} = \begin{bmatrix} 5 & 17 \ 11 & 17 \end{bmatrix}\]

Add Matrices B and C

Calculate \( B + C \) by adding the corresponding elements of matrices \( B \) and \( C \). \[B + C = \begin{bmatrix} -1 + 3 & 7 + 4 \ 8 + 1 & 4 + 9 \end{bmatrix} = \begin{bmatrix} 2 & 11 \ 9 & 13 \end{bmatrix}\]

Add Matrices A and B+C

Now, add matrix \( A \) to the result of \( B + C \) calculated in Step 3. \[A + (B + C) = \begin{bmatrix} 3 & 6 \ 2 & 4 \end{bmatrix} + \begin{bmatrix} 2 & 11 \ 9 & 13 \end{bmatrix} = \begin{bmatrix} 5 & 17 \ 11 & 17 \end{bmatrix}\]

Verify Equation (a)

Compare the results from Steps 2 and 4. Both \((A + B) + C\) and \(A + (B + C)\) yield \(\begin{bmatrix} 5 & 17 \ 11 & 17 \end{bmatrix}\). Therefore, the equation \((A + B) + C = A + (B + C)\) is verified.

Subtract Matrix C from A+B

Subtract matrix \( C \) from \( A + B \) calculated in Step 1.\[(A + B) - C = \begin{bmatrix} 2 & 13 \ 10 & 8 \end{bmatrix} - \begin{bmatrix} 3 & 4 \ 1 & 9 \end{bmatrix} = \begin{bmatrix} -1 & 9 \ 9 & -1 \end{bmatrix}\]

Subtract Matrix C from B

Calculate \( B - C \) by subtracting the corresponding elements of matrices \( C \) from \( B \).\[B - C = \begin{bmatrix} -1 & 7 \ 8 & 4 \end{bmatrix} - \begin{bmatrix} 3 & 4 \ 1 & 9 \end{bmatrix} = \begin{bmatrix} -4 & 3 \ 7 & -5 \end{bmatrix}\]

Add Matrices A and B-C

Add matrix \( A \) to the result of \( B - C \) calculated in Step 7.\[A + (B - C) = \begin{bmatrix} 3 & 6 \ 2 & 4 \end{bmatrix} + \begin{bmatrix} -4 & 3 \ 7 & -5 \end{bmatrix} = \begin{bmatrix} -1 & 9 \ 9 & -1 \end{bmatrix}\]

Verify Equation (b)

Compare the results from Steps 6 and 8. Both \((A + B) - C\) and \(A + (B - C)\) yield \(\begin{bmatrix} -1 & 9 \ 9 & -1 \end{bmatrix}\). Therefore, the equation \((A + B) - C = A + (B - C)\) is verified.

Key concepts

Matrix Addition

Matrix addition involves combining two matrices by adding their corresponding elements. Each matrix must have the same dimensions; that means they need to have the same number of rows and the same number of columns. This is because addition is done element-wise, meaning you add each element from one matrix to the corresponding element in the other matrix.

For example, if Matrix A has an element at the first row and first column, you add it only to the element at the same position in Matrix B. The formula looks like this:

• Given two matrices, A and B: \[ A + B = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix} \]

The matrices A, B, and C in the exercise all fit this requirement. To add matrices effectively, just line them up, and add each pair of corresponding numbers together.

Matrix Subtraction

Just like matrix addition, matrix subtraction involves subtracting corresponding elements of two matrices. Again, the matrices must have the same dimensions to perform element-wise subtraction.

If you have Matrix A and Matrix B, which are the same size, Matrix A - Matrix B is calculated by subtracting each element of B from the corresponding element of A. Here's how it looks:

• \[ A - B = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} - \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} - b_{11} & a_{12} - b_{12} \ a_{21} - b_{21} & a_{22} - b_{22} \end{bmatrix} \]

This operation allows us to compute differences between two matrices at each corresponding position, which can be useful for many applications in algebra.

Associative Property

The associative property in mathematics refers to the ability to group numbers or terms in an expression in different ways without changing the result. For matrix addition and subtraction, this property simplifies expressions by allowing operations to be done in any order.

For example, when matrices are involved:

• The associative property of addition states that \((A + B) + C = A + (B + C)\).
• The associative property of subtraction ensures that \((A - B) - C = A - (B + C)\). However, the subtraction of matrices is not commonly associative due to order significance. In matrix expressions, subtraction requires careful order of operations.

This property was shown in the solutions for both equations (a) and (b) where the parentheses could be switched without altering the outcome, demonstrating the ability to regroup the matrices as needed.

Commutative Property

The commutative property mainly applies to matrix addition, meaning that changing the order of the matrices does not affect their sum. If you have two matrices A and B:

• The commutative property states: \(A + B = B + A\).

This property simplifies the computation process because you can add matrices in any order you prefer.

In matrix subtraction, however, the commutative property does not hold. Switching the order will change the result because subtraction is order-sensitive, meaning \(A - B eq B - A\). It's important to remember which operation is being applied to ensure correct use of these properties. The commutative property allows for flexibility in matrix addition calculations, ensuring results remain consistent regardless of matrix positioning.