Problem 53 - Algebra 2

Question

Perform the indicated matrix operations. If the matrix does not exist, write impossible. $$ 4\left[\begin{array}{rrr}{3} & {4} & {-7} \\ {6} & {-9} & {-2} \\ {-3} & {1} & {3}\end{array}\right]-\left[\begin{array}{rrr}{-8} & {6} & {-4} \\ {-7} & {10} & {1} \\ {-2} & {1} & {5}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix is \( \begin{bmatrix} 20 & 10 & -24 \\ 31 & -46 & -9 \\ -10 & 3 & 7 \end{bmatrix} \).

Step 1: Multiply the First Matrix by 4

Multiply every element of the first matrix by 4.\[4 \begin{bmatrix} 3 & 4 & -7 \ 6 & -9 & -2 \ -3 & 1 & 3 \end{bmatrix} = \begin{bmatrix} 4 \times 3 & 4 \times 4 & 4 \times -7 \ 4 \times 6 & 4 \times -9 & 4 \times -2 \ 4 \times -3 & 4 \times 1 & 4 \times 3 \end{bmatrix} = \begin{bmatrix} 12 & 16 & -28 \ 24 & -36 & -8 \ -12 & 4 & 12 \end{bmatrix} \]

Step 2: Subtract the Second Matrix from the Resulting Matrix

Subtract the second matrix from the resulting matrix obtained in Step 1.\[\begin{bmatrix} 12 & 16 & -28 \ 24 & -36 & -8 \ -12 & 4 & 12 \end{bmatrix} - \begin{bmatrix} -8 & 6 & -4 \ -7 & 10 & 1 \ -2 & 1 & 5 \end{bmatrix} = \begin{bmatrix} 12 - (-8) & 16 - 6 & -28 - (-4) \ 24 - (-7) & -36 - 10 & -8 - 1 \ -12 - (-2) & 4 - 1 & 12 - 5 \end{bmatrix} = \begin{bmatrix} 20 & 10 & -24 \ 31 & -46 & -9 \ -10 & 3 & 7 \end{bmatrix}\]

Key Concepts

Matrix Subtraction

Matrix subtraction is a straightforward extension of regular number subtraction but applied to matrices. Unlike with numbers, you need to ensure that matrix subtraction is only performed on matrices of the same dimensions. This means both matrices must have the same number of rows and columns. With that in mind, here's how matrix subtraction works:

• Each element in the resulting matrix is derived by subtracting the corresponding elements from each matrix.
• For example, if we have two 3x3 matrices, the element in the first row and first column of the result is the result of subtracting the first row, first column element of the second matrix from that of the first matrix.
• It is important to remember to maintain the order of subtraction, as reversing it will yield different results.

Matrix subtraction keeps the resulting matrix's dimensions the same as the original matrices involved. Understanding these principles helps in ensuring the operations are done correctly in more complex scenarios.

Scalar Multiplication

Scalar multiplication is an important operation within matrix mathematics. It involves multiplying every entry of a matrix by a constant number known as a scalar. Unlike matrix multiplication which involves complex rules, scalar multiplication is simpler.

• The operation is straightforward: every single element of the matrix is multiplied by this scalar number.
• For instance, if you are given a matrix and a scalar like 4, multiplying this scalar by each element of the matrix results in another matrix of the same dimensions.
• This operation does not change the matrix shape, only the magnitude of its elements.

Scalar multiplication is incredibly useful because it's one of the core tools in transforming matrices. This operation is often used as a preliminary step, as seen in our original exercise, where the matrix was first adjusted with scalar multiplication before other operations like subtraction.

Algebraic Expressions

In mathematics, algebraic expressions are a vital part of representing and solving problems. This applies to matrix operations as well. Algebraic expressions in matrix operations typically involve combining various operations like addition, subtraction, and scalar multiplication within the same problem.

• An algebraic expression can include constants, variables, and operations within matrices.
• In the context of matrix operations, algebraic expressions help in representing how we modify matrices through combinations of the discussed operations.
• Real-life applications involve using algebraic expressions to describe systems of equations and perform necessary calculations for solutions in physics and engineering.

Successful manipulation of matrix algebraic expressions can enhance the ability to solve complex mathematical problems involving matrices more efficiently.