Problem 40 - Algebra and Trigonometry

Question

Perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} {4} & {0} \\ {-3} & {5} \\ {0} & {1} \end{array}\right] \quad B=\left[\begin{array}{rr} {5} & {1} \\ {-2} & {-2} \end{array}\right] \quad C=\left[\begin{array}{rr} {1} & {-1} \\ {-1} & {1} \end{array}\right] $$ $$ A(B+C) $$

Step-by-Step Solution

Answer

The result of A(B+C) is \( \left[\begin{array}{cc} 24 & 0 \ -33 & -5 \ -3 & -1 \end{array}\right] \).

Step 1: Perform Matrix Addition first

The parentheses indicate that addition should be performed first. Since B & C are both \(2 \times 2\) matrices, addition is possible. Add the corresponding elements of matrices B & C.. This will give: \( \left[\begin{array}{cc} 5 + 1 & 1 + -1 \ -2 + -1 & -2 + 1 \end{array}\right] = \left[\begin{array}{cc} 6 & 0 \ -3 & -1 \end{array}\right] \). Let's call this resultant matrix D.

Step 2: Perform Matrix Multiplication

Now multiply matrix A by matrix D. For matrix multiplication, the element at the i-th row and j-th column of the resultant matrix is obtained by multiplying each element of the i-th row of the first matrix by the corresponding element of the j-th column of the second matrix, and then adding them. However, remember to check if the operation is defined. Since A is a \(3 \times 2\) matrix and D is a \(2 \times 2\) matrix, it's possible to perform the multiplication (number of columns of the first matrix equal the number of rows of the second). Upon performing the multiplication, we get: \( \left[\begin{array}{cc} 4 \times 6 + 0 \times -3 & 4 \times 0 + 0 \times -1 \ -3 \times 6 + 5 \times -3 & -3 \times 0 + 5 \times -1 \ 0 \times 6 + 1 \times -3 & 0 \times 0 + 1 \times -1 \end{array}\right] = \left[\begin{array}{cc} 24 & 0 \ -33 & -5 \ -3 & -1 \end{array}\right] \).

Step 3: Final Result

So, the matrix A(B+C) equals \( \left[\begin{array}{cc} 24 & 0 \ -33 & -5 \ -3 & -1 \end{array}\right] \).

Key Concepts

Matrix Addition

Matrix addition is one of the fundamental operations in linear algebra. It involves combining two matrices by adding their corresponding elements together. For this operation to be possible, both matrices must be of the same size, meaning they have the same number of rows and columns.

In the given exercise, the matrices B and C, both being 2x2 matrices, were added to get matrix D. Each element of matrix B was added to the corresponding element of matrix C. For instance, the top-left elements (5 from B and 1 from C) were added to result in 6 for the top-left position in matrix D. Similarly, other elements were combined to form the complete resultant matrix D.

Matrix Multiplication

Matrix multiplication is another core operation that involves combining matrices in a more complex manner than addition. When multiplying two matrices, the element in the i-th row and j-th column of the product matrix is calculated by taking the dot product of the i-th row of the first matrix with the j-th column of the second matrix.

In our example, matrix A was a 3x2 matrix and matrix D (resulting from B+C) was a 2x2 matrix. The matrix multiplication was possible because the number of columns in A was equal to the number of rows in D. The entries in the product matrix were calculated accordingly. For example, the first element was computed by multiplying 4 (from the first row of A) by 6 (from the first column of D) and adding the product to the result of multiplying 0 (from the first row of A) by -3 (from the second column of D), yielding 24.

Matrices

Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. Each individual number in a matrix is called an element. Matrices are often used in mathematics to compactly represent and operate on sets of equations or data.

The size or dimension of a matrix is defined by the number of rows and columns it contains. In the exercise discussed, matrix A is a 3x2 matrix, meaning it has 3 rows and 2 columns. The matrices B and C are 2x2 matrices. The operations on matrices, including addition and multiplication as shown in the exercise, follow specific rules based on the dimensions of the matrices involved. Matrices are powerful mathematical tools used in various fields including physics, engineering, computer science, and economics for solving systems of linear equations, transforming geometric data, and modeling different types of problems.