Problem 38 - 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 \(k(A+B)=k A+k B\) for any real number \(k\).

Step-by-Step Solution

Answer

The equality \(k(A + B) = kA + kB\) holds as shown by performing matrix addition and scalar multiplication.

Step 1: Identify Matrix Properties

Understand that matrices can be added by adding their corresponding elements and that scalar multiplication of a matrix involves multiplying each element of the matrix by the scalar.

Step 2: Express A + B

Calculate the sum of matrices A and B. This involves adding their corresponding elements.\[A + B = \begin{bmatrix}a_{11} + b_{11} & a_{12} + b_{12} \a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}\]

Step 3: Apply Scalar Multiplication to A + B

Multiply each element of the matrix \(A + B\) by the scalar \(k\).\[k(A + B) = \begin{bmatrix}k(a_{11} + b_{11}) & k(a_{12} + b_{12}) \k(a_{21} + b_{21}) & k(a_{22} + b_{22}) \end{bmatrix}\]

Step 4: Distribute the Scalar Over A and B

Apply scalar multiplication to matrices \(A\) and \(B\) separately.\[ kA = \begin{bmatrix} ka_{11} & ka_{12} \ka_{21} & ka_{22} \end{bmatrix}, \ kB = \begin{bmatrix} kb_{11} & kb_{12} \kb_{21} & kb_{22} \end{bmatrix} \]

Step 5: Add kA and kB

Add the matrices \(kA\) and \(kB\) by adding their corresponding elements.\[kA + kB = \begin{bmatrix}ka_{11} + kb_{11} & ka_{12} + kb_{12} \ka_{21} + kb_{21} & ka_{22} + kb_{22} \end{bmatrix}\]

Step 6: Verify Equality

Note that the results from Step 3 and Step 5 are the same, hence \(k(A+B) = kA + kB\). This confirms the property of scalar multiplication over matrix addition.

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. To perform matrix addition, both matrices must have the same dimensions. This means each matrix should have the same number of rows and columns. The process is simple and systematic:

• Align the matrices and add each element in a position of one matrix to the element in the same position of the other matrix.
• For example, if you have Matrix A as \( \begin{aligned} A = \left[\begin{array}{cc} a_{11} & a_{12} \ a_{21} & a_{22} \end{array}\right] \end{aligned} \) and Matrix B as \( \begin{aligned} B = \left[\begin{array}{cc} b_{11} & b_{12} \ b_{21} & b_{22} \end{array}\right] \end{aligned} \), then their sum is \(A + B = \left[ \begin{array}{cc} a_{11}+b_{11} & a_{12}+b_{12} \ a_{21}+b_{21} & a_{22}+b_{22} \end{array}\right] \).

Matrix addition is commutative and associative, which means the order of addition does not affect the result. This operation helps in solving systems of equations and connecting various mathematical models.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a scalar number. This operation scales the entire matrix by the given scalar and retains the structure of the matrix. Here's how it works:

• Given a matrix, each element is multiplied by the scalar individually.
• For example, if matrix A is \( \begin{aligned} A = \left[\begin{array}{cc} a_{11} & a_{12} \ a_{21} & a_{22} \end{array}\right] \end{aligned} \) and the scalar is \(k\), then \(kA = \left[ \begin{array}{cc} ka_{11} & ka_{12} \ ka_{21} & ka_{22} \end{array}\right] \).

Scalar multiplication is an important concept because it illustrates how matrices can be transformed or scaled in a variety of applications, such as physics and computer graphics. It’s straightforward but crucial for understanding more complex matrix operations.

Distributive Property

The distributive property is a key concept in algebra that also applies to matrices. It explains how multiplication can be distributed over addition. When applied to matrices, it shows that multiplying a scalar with a matrix sum yields the same result as multiplying the scalar with each individual matrix and then adding them. The formula is:

• If \(A\) and \(B\) are matrices of the same size, and \(k\) is a scalar, then \(k(A + B) = kA + kB\).

Here's the breakdown:

• First, add matrices \(A\) and \(B\) to get \(A + B\).
• Multiply the result by \(k\), applying the scalar to each element.
• Alternatively, multiply each matrix \(A\) and \(B\) by \(k\) separately, then add the results: \(kA + kB\).

This property makes matrix equations more manageable and is a practical tool in solving algebraic problems involving matrices.

Matrices

Matrices are rectangular arrays of numbers arranged in rows and columns. They are a central object in mathematics and have applications across various fields.

• The format of a matrix is basically a grid, where each entry is called an element.
• Matrices are typically denoted by uppercase letters like \(A\), \(B\), or \(C\).

Matrices serve several purposes:

• They represent systems of linear equations compactly.
• In computer graphics, they help in transforming geometric data.
• In statistics, they organize data sets efficiently.

The size of a matrix is defined by its dimensions, which indicate the number of rows and columns. Understanding the structure and arithmetic of matrices is essential for navigating more advanced topics that hinge on these mathematical objects.