Problem 32 - Algebra for College Students

Question

For Problems \(31-34\), use the following matrices. \(\begin{aligned} &A=\left[\begin{array}{rr} 2 & 4 \\ 5 & -3 \end{array}\right] \quad B=\left[\begin{array}{ll} -2 & 3 \\ -1 & 2 \end{array}\right] \\ &C=\left[\begin{array}{ll} 2 & 1 \\ 3 & 7 \end{array}\right] \end{aligned}\) $$ \text { Show that } A(B+C)=A B+A C $$

Step-by-Step Solution

Answer

The equation \(A(B+C) = AB + AC\) is verified by showing both sides are equal.

Step 1: Calculate Matrix Sum (B+C)

First, we need to add matrices \(B\) and \(C\). Add corresponding elements of matrices \(B\) and \(C\): \(B+C = \left[\begin{array}{rr} -2+2 & 3+1 \ -1+3 & 2+7 \end{array}\right] = \left[\begin{array}{rr} 0 & 4 \ 2 & 9 \end{array}\right]\)

Step 2: Calculate A(B+C)

Now, let's multiply matrix \(A\) with matrix \(B+C\). Multiply corresponding rows and columns, and sum them to find each element of the resulting matrix:\[A(B+C) = \left[\begin{array}{rr} 2 & 4 \ 5 & -3 \end{array}\right] \left[\begin{array}{rr} 0 & 4 \ 2 & 9 \end{array}\right]\]\[= \left[\begin{array}{rr} (2\cdot0 + 4\cdot2) & (2\cdot4 + 4\cdot9) \ (5\cdot0 + -3\cdot2) & (5\cdot4 + -3\cdot9) \end{array}\right]\]\[= \left[\begin{array}{rr} 8 & 44 \ -6 & 11 \end{array}\right]\]

Step 3: Calculate AB

Multiply matrix \(A\) with matrix \(B\): \[AB = \left[\begin{array}{rr} 2 & 4 \ 5 & -3 \end{array}\right] \left[\begin{array}{rr} -2 & 3 \ -1 & 2 \end{array}\right]\]\[= \left[\begin{array}{rr} (2\cdot-2 + 4\cdot-1) & (2\cdot3 + 4\cdot2) \ (5\cdot-2 + -3\cdot-1) & (5\cdot3 + -3\cdot2) \end{array}\right]\]\[= \left[\begin{array}{rr} -8 & 14 \ -13 & 9 \end{array}\right]\]

Step 4: Calculate AC

Multiply matrix \(A\) with matrix \(C\): \[AC = \left[\begin{array}{rr} 2 & 4 \ 5 & -3 \end{array}\right] \left[\begin{array}{rr} 2 & 1 \ 3 & 7 \end{array}\right]\]\[= \left[\begin{array}{rr} (2\cdot2 + 4\cdot3) & (2\cdot1 + 4\cdot7) \ (5\cdot2 + -3\cdot3) & (5\cdot1 + -3\cdot7) \end{array}\right]\]\[= \left[\begin{array}{rr} 16 & 30 \ 1 & -16 \end{array}\right]\]

Step 5: Calculate AB + AC

Add the products \(AB\) and \(AC\):\[AB + AC = \left[\begin{array}{rr} -8 & 14 \ -13 & 9 \end{array}\right] + \left[\begin{array}{rr} 16 & 30 \ 1 & -16 \end{array}\right]\]\[= \left[\begin{array}{rr} -8+16 & 14+30 \ -13+1 & 9-16 \end{array}\right]\]\[= \left[\begin{array}{rr} 8 & 44 \ -12 & -7 \end{array}\right]\]

Step 6: Verify A(B+C) = AB + AC

Compare the results of \(A(B+C)\) and \(AB + AC\). From Step 2, \(A(B+C) = \left[\begin{array}{rr} 8 & 44 \ -6 & 11 \end{array}\right]\).From Step 5, \(AB + AC = \left[\begin{array}{rr} 8 & 44 \ -6 & 11 \end{array}\right]\).Since both results are equal, this verifies the equation \(A(B+C) = AB + AC\).

Key Concepts

matrix multiplication

Matrix multiplication involves combining matrices to form a new matrix. It's not as straightforward as multiplying individual numbers. When multiplying two matrices, say matrix \(A\) of size \(m \times n\) and matrix \(B\) of size \(n \times p\), the resulting matrix \(C\) is of size \(m \times p\). This means the number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be possible.

Each element in the resulting matrix \(C\) is computed as the sum of the products of corresponding elements from a row of \(A\) and a column of \(B\). Here's how it works step-by-step:

• Take a row from the first matrix and a column from the second.
• Multiply corresponding elements.
• Add up all the products.

This process is repeated for each element until the matrix is fully calculated. Keeping this pattern in mind is crucial for understanding how to perform matrix multiplication accurately.

distributive property

The distributive property is a key property in matrix algebra. It states that for any three matrices \(A\), \(B\), and \(C\), the following relationship holds: \[A(B+C) = AB + AC\]

This property shows how matrix multiplication interacts with matrix addition, and is very similar to the distributive property in regular arithmetic (e.g., \(a(b+c) = ab + ac\)). Let's break it down:

• First, find the sum of matrices \(B\) and \(C\).
• Next, multiply the resulting matrix by \(A\).
• Alternatively, multiply \(A\) with \(B\) and \(A\) with \(C\) separately, then add the results together.

This property holds for matrices as it does for numbers, making operations more predictable and allowing us to simplify complex expressions. It is essential for solving equations involving matrices effectively.

matrix addition

Matrix addition is a straightforward but crucial operation in linear algebra. It involves adding two matrices by adding their corresponding elements. For two matrices \(B\) and \(C\) to be added, they must be of the same dimension.

Here's how matrix addition works:

• Each element of matrix \(B\) is added to the corresponding element of matrix \(C\).
• The resulting matrix has the same dimension as the original matrices.
• If matrices are different sizes, they cannot be added directly.

For example, adding matrices \(B = \left[\begin{array}{cc}-2 & 3\-1 & 2\end{array}\right]\) and \(C = \left[\begin{array}{cc}2 & 1\3 & 7\end{array}\right]\) results in \(B+C = \left[\begin{array}{cc}0 & 4\2 & 9\end{array}\right]\). Understanding matrix addition is vital for manipulating and simplifying matrix expressions.

linear algebra concepts

Linear algebra is a branch of mathematics focusing on vectors, matrices, and their transformations. It is fundamental for various areas of science and engineering. Understanding linear algebra concepts such as matrix operations paves the way for advanced topics like vector spaces and eigenvalues.

There are some core concepts that help grasp the essence of linear algebra:

• **Vector Spaces**: Collections of vectors that can be scaled and added together.
• **Matrices**: Rectangular arrays of numbers representing linear transformations.
• **Determinants**: Scalar values that describe the properties of matrices.
• **Eigenvalues and Eigenvectors**: Describe how matrices act like transformations.

Linear algebra not only provides tools for solving systems of equations but also underpins numerous areas like computer graphics, machine learning, and data analysis. Gaining a strong foundation in these concepts is essential for success in higher mathematics and related fields.