Problem 34 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

Equation \((3+2)A=3A+2A\) is confirmed by matrix operations.

Step 1: Simplify the Scalar Addition

First, simplify the expression on the left-hand side of the equation. Calculate the sum of the scalars: \(3 + 2 = 5\)

Step 2: Scalar Multiplication on Left-hand Side

Perform scalar multiplication on the left side of the equation. Multiply each element of matrix \(A\) by the scalar \(5\):\[5A = 5 \times \begin{bmatrix} 2 & 4 \ 5 & -3 \end{bmatrix} = \begin{bmatrix} 5 \times 2 & 5 \times 4 \ 5 \times 5 & 5 \times -3 \end{bmatrix} = \begin{bmatrix} 10 & 20 \ 25 & -15 \end{bmatrix}\]

Step 3: Distribute Scalars on Right-hand Side

Distribute the scalars on the right-hand side. First, multiply \(3\) and \(2\) by matrix \(A\) separately:\(3A = 3 \times \begin{bmatrix} 2 & 4 \ 5 & -3 \end{bmatrix} = \begin{bmatrix} 6 & 12 \ 15 & -9 \end{bmatrix}\)\(2A = 2 \times \begin{bmatrix} 2 & 4 \ 5 & -3 \end{bmatrix} = \begin{bmatrix} 4 & 8 \ 10 & -6 \end{bmatrix}\)

Step 4: Add Results from Right-hand Side

Add the matrices from the results of scalar multiplications:\[3A + 2A = \begin{bmatrix} 6 & 12 \ 15 & -9 \end{bmatrix} + \begin{bmatrix} 4 & 8 \ 10 & -6 \end{bmatrix}\]Perform the addition by adding corresponding elements:\[= \begin{bmatrix} 6+4 & 12+8 \ 15+10 & -9+(-6) \end{bmatrix} = \begin{bmatrix} 10 & 20 \ 25 & -15 \end{bmatrix}\]

Step 5: Compare Both Sides

Compare the results from the left-hand side and the right-hand side of the equation. We see:Left-hand side: \(\begin{bmatrix} 10 & 20 \ 25 & -15 \end{bmatrix}\)Right-hand side: \(\begin{bmatrix} 10 & 20 \ 25 & -15 \end{bmatrix}\)Since both are equal, the equation \((3+2)A = 3A + 2A\) holds true.

Key Concepts

Matrix Addition

Matrix Addition is an essential idea in mathematics where two matrices of the same size are combined by adding their corresponding elements. If you have matrices with the same dimensions, you line up the elements and simply add the numbers from each set of matrices that are in the same position. For example, if you have two matrices:

• Matrix X = \[ \begin{bmatrix} x_1 & x_2 \ x_3 & x_4 \end{bmatrix} \]
• Matrix Y = \[ \begin{bmatrix} y_1 & y_2 \ y_3 & y_4 \end{bmatrix} \]

Matrix X + Y would be calculated as:\[ \begin{bmatrix} x_1 + y_1 & x_2 + y_2 \ x_3 + y_3 & x_4 + y_4 \end{bmatrix} \].
For matrix addition to work, both matrices must have the same dimensions. This means the number of rows and columns in each matrix must match perfectly. This is important when performing any matrix operations involving addition.

Scalar Multiplication Property

Scalar Multiplication is an operation where every element of a matrix is multiplied by a single numerical value (called a scalar). This property is vital because it allows us to easily scale the entire matrix by one number. If you have a scalar, let's say 'k' and a matrix:

• Matrix Z = \[ \begin{bmatrix} z_1 & z_2 \z_3 & z_4 \end{bmatrix} \]

Then scalar multiplication of 'k' and Matrix Z results in:\[ k \times \begin{bmatrix} z_1 & z_2 \ z_3 & z_4 \end{bmatrix} = \begin{bmatrix} k \cdot z_1 & k \cdot z_2 \ k \cdot z_3 & k \cdot z_4 \end{bmatrix} \].
The key takeaway is the simplification it offers: scaling all elements by the same amount efficiently without worrying about individual values. It is an essential skill for handling more complex matrix operations.

Matrix Operations

Matrix operations include a range of activities such as addition, subtraction, multiplication, and scalar multiplication, formed around matrix fundamentals. Follow these rules:

• For addition and subtraction, matrices must have the same dimensions.
• For multiplication, the number of columns in the first matrix must equal the number of rows in the second.
• Scalar multiplication involves multiplying each matrix element by a scalar value.

Understanding these operations is critical as they pave the way for exploring more advanced mathematical concepts, such as transformations and equations in algebra and geometry. Practicing these basics ensures you can effortlessly handle matrices in any given context.

Algebraic Proofs

Algebraic proofs involve demonstrating that a mathematical statement is true through logical reasoning and using known definitions and properties. In our exercise, the equation \((3+2)A = 3A + 2A\) depends on such reasoning.
To prove it:

• First, simplify \(3 + 2\) to obtain \(5A\).
• Then, apply scalar multiplication to matrix \(A\) with a scalar of \(5\).
• For the right side, distribute the scalar across \(A\) to achieve \(3A + 2A\).

Using matrix addition, combine \(3A\) and \(2A\) to match \(5A\).
This approach not only confirms the equation but strengthens our understanding of how scalar multiplication and addition interact. Such exercises illustrate how algebraic operations apply beyond typical number manipulation, enhancing comprehension of broader mathematical concepts.