Problem 3 - Algebra 2

Question

Use matrices \(A, B, C,\) and \(D .\) Find each product, sum, or difference. $$A=\left[\begin{array}{rr}{3} & {4} \\ {6} & {-2} \\ {1} & {0}\end{array}\right] \quad B=\left[\begin{array}{rr}{-3} & {1} \\ {2} & {-4} \\\ {-1} & {5}\end{array}\right] \quad C=\left[\begin{array}{rr}{1} & {2} \\\ {-3} & {1}\end{array}\right] \quad D=\left[\begin{array}{ll}{5} & {1} \\ {0} & {2}\end{array}\right]$$ $$ -3 C $$

Step-by-Step Solution

Answer

The product of -3 and matrix C is \[ -3C=\left[\begin{array}{cc} -3 & -6 \ 9 & -3 \end{array}\right]\].

Step 1: Understand Matrix Scalar Multiplication

Matrix scalar multiplication involves multiplying every entry of a matrix by a scalar (a real number). In this case, we are multiplying every entry of matrix C by -3.

Step 2: Multiply each element of Matrix C by -3

To find -3C, multiply each element of matrix C by -3. This is done element-wise, meaning each entry in the resulting matrix is the product of -3 and the corresponding entry in C.

Step 3: Write the resulting matrix

After multiplying each element of C by -3, write down the resulting matrix to complete the operation.

Key Concepts

Matrices in Algebra

Matrices, fundamental to the field of algebra, serve as arrangements of numbers, expressions, or symbols grouped in rows and columns to form a rectangular array. Understanding how to manipulate these structured arrays is crucial for solving various algebraic problems, especially those set in higher dimensions. In algebra, matrices are employed for representing and solving systems of linear equations, transforming geometric objects, and encoding data in areas such as computer graphics and statistics. The essence of matrix operations, like addition, subtraction, and multiplication, reflects the significance of this tool in simplifying complex problems into more manageable calculations.

Matrix Arithmetic

Just like with single numbers, matrices can be added, subtracted, and multiplied through specific rules that preserve the structure of these mathematical objects. Matrix arithmetic is the collection of these operations. However, these are not commutative in nature, meaning the order of operation matters, especially with multiplication. For addition and subtraction, matrices must possess the same dimensions, and the operations are performed element-wise. On the other hand, multiplication demands more attention to the matrices' sizes, as certain dimensions must conform to allow the operation. It is this framework that enables matrices to solve complex linear mappings and transformations.

Element-wise Multiplication

Element-wise multiplication, also known as the Hadamard product, is a binary operation that takes two matrices of the same dimensions and produces another matrix where each element \(i,j\) is the product of the elements \(i,j\) of the original matrices. Nonetheless, it is not to be confused with matrix scalar multiplication or the matrix product. In matrix scalar multiplication, as showcased in the exercise with the matrix C and scalar -3, each element of the matrix is multiplied by the scalar, resulting in each value of the original matrix being scaled by this number. This operation is fundamental in altering the scaling of vectors and in transformation operations in geometry and physics.