Problem 42 - Algebra for College Students

Question

For Problems \(35-43\), use the following matrices. \( \begin{aligned} A &=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] & B &=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right] \\ C &=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] & O &=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \end{aligned} \) Show that \(A(B+C)=A B+A C\).

Step-by-Step Solution

Answer

Matrix multiplication distributes over addition, showing \(A(B+C) = AB + AC\).

Step 1: Matrix Addition

First, add matrices \(B\) and \(C\). The addition of two matrices involves adding the corresponding elements. Thus, \(B + C\) results in the matrix: \(\begin{bmatrix} b_{11} + c_{11} & b_{12} + c_{12} \ b_{21} + c_{21} & b_{22} + c_{22} \end{bmatrix}\).

Step 2: Matrix Multiplication with A

Next, multiply matrix \(A\) by the resultant matrix \(B + C\). Using the rule of matrix multiplication, \(A(B + C)\) results in the matrix: \(\begin{bmatrix} a_{11}(b_{11} + c_{11}) + a_{12}(b_{21} + c_{21}) & a_{11}(b_{12} + c_{12}) + a_{12}(b_{22} + c_{22}) \ a_{21}(b_{11} + c_{11}) + a_{22}(b_{21} + c_{21}) & a_{21}(b_{12} + c_{12}) + a_{22}(b_{22} + c_{22}) \end{bmatrix}\).

Step 3: Multiply A with B

Calculate the product of \(A\) and \(B\) separately: \(AB = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix}\).

Step 4: Multiply A with C

Next, calculate the product of \(A\) and \(C\): \(AC = \begin{bmatrix} a_{11}c_{11} + a_{12}c_{21} & a_{11}c_{12} + a_{12}c_{22} \ a_{21}c_{11} + a_{22}c_{21} & a_{21}c_{12} + a_{22}c_{22} \end{bmatrix}\).

Step 5: Sum the Products

Add the matrices \(AB\) and \(AC\). Summing corresponding elements gives: \(AB + AC = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} + a_{11}c_{11} + a_{12}c_{21} & a_{11}b_{12} + a_{12}b_{22} + a_{11}c_{12} + a_{12}c_{22} \ a_{21}b_{11} + a_{22}b_{21} + a_{21}c_{11} + a_{22}c_{21} & a_{21}b_{12} + a_{22}b_{22} + a_{21}c_{12} + a_{22}c_{22} \end{bmatrix}\).

Step 6: Verify Equality

Compare the matrix \(A(B+C)\) from Step 2 with the matrix \(AB + AC\) from Step 5. They are identical, confirming that \(A(B+C) = AB + AC\).

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation where you sum corresponding elements of two matrices. To perform matrix addition successfully, ensure that both matrices have the same dimensions. Each element from the first matrix is added to the element in the same position in the second matrix. Given matrices like: \[ B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}, \ C = \begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{bmatrix} \] You calculate \( B + C \) by summing corresponding elements: * \( b_{11} + c_{11} \)* \( b_{12} + c_{12} \)* \( b_{21} + c_{21} \)* \( b_{22} + c_{22} \) This results in a new matrix: \[ \begin{bmatrix} b_{11} + c_{11} & b_{12} + c_{12} \ b_{21} + c_{21} & b_{22} + c_{22} \end{bmatrix} \] By ensuring matching dimensions, matrix addition becomes straightforward and is an essential building block for more complex operations.

Matrix Multiplication

Matrix multiplication involves multiplying two matrices to produce a third matrix. Unlike addition, multiplication has specific rules and requires the number of columns in the first matrix to match the number of rows in the second matrix. Suppose matrix \( A \) is defined as: \[ A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \] When multiplying \( A \) by another matrix \( B \), the resulting matrix's elements are calculated using specific formulas: * \( a_{11}b_{11} + a_{12}b_{21} \) becomes the first element* \( a_{11}b_{12} + a_{12}b_{22} \) becomes the second element* Similar calculations are done for the remaining elements of the resulting matrix It's important to perform each operation meticulously to ensure accuracy. Matrix multiplication is crucial for handling transformations in many scientific fields and understanding how combined data sets interact.

Matrices

Matrices are powerful mathematical tools represented as rectangular arrays of numbers, symbols, or expressions. They are an essential part of linear algebra and have applications across all areas of science and engineering. Each entry in a matrix is labeled based on its row and column position. Matrices can be of varying sizes and configurations, such as 2x2 or 3x3, based on the number of rows and columns. The operations that involve matrices include addition, subtraction, and multiplication, along with finding determinants and inverses. In practical terms, matrices handle a lot of information compactly. They are used in various complex operations from computer graphics to systems of equations solving. Always remember: * Matrices must align properly to perform certain operations like matrix multiplication. * The size of matrices dictates potential operations — a simple check can prevent errors. Their utility makes matrices indispensable for mathematical modeling and problem-solving.