Problem 7
Question
Use matrices \(A, B, C,\) and \(D .\) Find each product, sum, or difference. $$A=\left[\begin{array}{rr}{3} & {4} \\ {6} & {-2} \\ {1} & {0}\end{array}\right] \quad B=\left[\begin{array}{rr}{-3} & {1} \\ {2} & {-4} \\\ {-1} & {5}\end{array}\right] \quad C=\left[\begin{array}{rr}{1} & {2} \\\ {-3} & {1}\end{array}\right] \quad D=\left[\begin{array}{ll}{5} & {1} \\ {0} & {2}\end{array}\right]$$ $$ 4 C+3 D $$
Step-by-Step Solution
Verified Answer
\left[\begin{array}{rr}{19} & {11} \ {-12} & {10}\end{array}\right].
1Step 1 - Calculate 4C
Multiply each element of matrix C by 4: \( 4C = 4 \times \left[\begin{array}{rr}{1} & {2} \ {-3} & {1}\end{array}\right] = \left[\begin{array}{rr}{4 \times 1} & {4 \times 2} \ {4 \times (-3)} & {4 \times 1}\end{array}\right] = \left[\begin{array}{rr}{4} & {8} \ {-12} & {4}\end{array}\right] \).
2Step 2 - Calculate 3D
Multiply each element of matrix D by 3: \( 3D = 3 \times \left[\begin{array}{ll}{5} & {1} \ {0} & {2}\end{array}\right] = \left[\begin{array}{ll}{3 \times 5} & {3 \times 1} \ {3 \times 0} & {3 \times 2}\end{array}\right] = \left[\begin{array}{ll}{15} & {3} \ {0} & {6}\end{array}\right] \).
3Step 3 - Add the resulting matrices
Add the matrices from Step 1 and Step 2 together: \( 4C + 3D = \left[\begin{array}{rr}{4} & {8} \ {-12} & {4}\end{array}\right] + \left[\begin{array}{ll}{15} & {3} \ {0} & {6}\end{array}\right] = \left[\begin{array}{rr}{4 + 15} & {8 + 3} \ {-12 + 0} & {4 + 6}\end{array}\right] = \left[\begin{array}{rr}{19} & {11} \ {-12} & {10}\end{array}\right] \).
Key Concepts
Matrix AdditionMatrix Scalar MultiplicationMatrix AlgebraMatrices in Algebra
Matrix Addition
Matrix addition is a fundamental operation in matrix algebra which involves adding two matrices by adding the corresponding elements together. For matrices to be added, they must be of the same size, meaning they have the same number of rows and columns.
To illustrate, if we have two matrices, matrix \(A\) and matrix \(B\), the sum \(A + B\) is found by adding each element \(a_{ij}\) from \(A\) to the corresponding element \(b_{ij}\) from \(B\), resulting in a new matrix with elements \(a_{ij} + b_{ij}\). The operation is performed element-wise as follows:
\[ (A + B) = \left[\begin{array}{ccc} a_{11} + b_{11} & \cdots & a_{1n} + b_{1n} \ \vdots & \ddots & \vdots \ a_{m1} + b_{m1} & \cdots & a_{mn} + b_{mn} \end{array}\right] \]
Using the provided exercise, the addition of \(4C + 3D\) demonstrates matrix addition, with each corresponding element of the scaled matrices \(4C\) and \(3D\) being summed to produce the resulting matrix.
To illustrate, if we have two matrices, matrix \(A\) and matrix \(B\), the sum \(A + B\) is found by adding each element \(a_{ij}\) from \(A\) to the corresponding element \(b_{ij}\) from \(B\), resulting in a new matrix with elements \(a_{ij} + b_{ij}\). The operation is performed element-wise as follows:
\[ (A + B) = \left[\begin{array}{ccc} a_{11} + b_{11} & \cdots & a_{1n} + b_{1n} \ \vdots & \ddots & \vdots \ a_{m1} + b_{m1} & \cdots & a_{mn} + b_{mn} \end{array}\right] \]
Using the provided exercise, the addition of \(4C + 3D\) demonstrates matrix addition, with each corresponding element of the scaled matrices \(4C\) and \(3D\) being summed to produce the resulting matrix.
Matrix Scalar Multiplication
Matrix scalar multiplication is the process of multiplying every element of a matrix by the same scalar value. This operation affects the magnitude of matrix elements but does not change the size of the matrix.
The scalar multiplication of a matrix \(A\) by a scalar \(k\) is calculated as follows:
\[ kA = k \times \left[\begin{array}{ccc} a_{11} & \cdots & a_{1n} \ \vdots & \ddots & \vdots \ a_{m1} & \cdots & a_{mn} \end{array}\right] = \left[\begin{array}{ccc} k \times a_{11} & \cdots & k \times a_{1n} \ \vdots & \ddots & \vdots \ k \times a_{m1} & \cdots & k \times a_{mn} \end{array}\right] \]
As demonstrated in the given solution, when matrix \(C\) is multiplied by the scalar 4, every element within \(C\) is multiplied by 4. Similarly, every element in matrix \(D\) is multiplied by the scalar 3. This yields the matrices known as \(4C\) and \(3D\), which we can then add together in the matrix addition step.
The scalar multiplication of a matrix \(A\) by a scalar \(k\) is calculated as follows:
\[ kA = k \times \left[\begin{array}{ccc} a_{11} & \cdots & a_{1n} \ \vdots & \ddots & \vdots \ a_{m1} & \cdots & a_{mn} \end{array}\right] = \left[\begin{array}{ccc} k \times a_{11} & \cdots & k \times a_{1n} \ \vdots & \ddots & \vdots \ k \times a_{m1} & \cdots & k \times a_{mn} \end{array}\right] \]
As demonstrated in the given solution, when matrix \(C\) is multiplied by the scalar 4, every element within \(C\) is multiplied by 4. Similarly, every element in matrix \(D\) is multiplied by the scalar 3. This yields the matrices known as \(4C\) and \(3D\), which we can then add together in the matrix addition step.
Matrix Algebra
Matrix algebra encompasses various operations that can be performed on matrices, including addition, scalar multiplication, matrix multiplication, and transposition, among others. It's a branch of mathematics that deals with the study of matrices and the rules for manipulating these arrays of numbers.
Matrix algebra is particularly useful in solving systems of linear equations, transforming geometric figures, and representing and working with linear transformations. The rules of matrix algebra resemble arithmetic rules, but with notable exceptions, such as matrix multiplication not being commutative.
In the provided exercise, we use the concepts of matrix algebra to perform scalar multiplication followed by matrix addition, showcasing the systematic and rule-governed nature of matrix operations. Understanding how to correctly apply these operations is essential for working in various fields of science and engineering.
Matrix algebra is particularly useful in solving systems of linear equations, transforming geometric figures, and representing and working with linear transformations. The rules of matrix algebra resemble arithmetic rules, but with notable exceptions, such as matrix multiplication not being commutative.
In the provided exercise, we use the concepts of matrix algebra to perform scalar multiplication followed by matrix addition, showcasing the systematic and rule-governed nature of matrix operations. Understanding how to correctly apply these operations is essential for working in various fields of science and engineering.
Matrices in Algebra
Matrices play a crucial role in algebra, serving as a powerful tool for representing complex problems in a concise and manageable format. They are particularly effective for representing and solving linear equations, vectors, and linear transformations.
Moreover, matrices provide a framework for dealing with data structures in computer science, they assist in encryption algorithms for cybersecurity, and they play an integral part in graphics and image processing. The utility of matrices extends even further into fields like physics, economics, statistics, and any area involving large amounts of data or systems behavior modeling.
With exercises like the one given, learning how matrices can be manipulated through operations such as addition and scalar multiplication allows students to appreciate the vast applicability of matrix theory across a spectrum of disciplines.
Moreover, matrices provide a framework for dealing with data structures in computer science, they assist in encryption algorithms for cybersecurity, and they play an integral part in graphics and image processing. The utility of matrices extends even further into fields like physics, economics, statistics, and any area involving large amounts of data or systems behavior modeling.
With exercises like the one given, learning how matrices can be manipulated through operations such as addition and scalar multiplication allows students to appreciate the vast applicability of matrix theory across a spectrum of disciplines.
Other exercises in this chapter
Problem 7
Use a graphing calculator to evaluate the determinant of each \(3 \times 3\) matrix. $$ \left[\begin{array}{ccc}{12.2} & {13.3} & {9} \\ {1} & {-4} & {-17} \\\
View solution Problem 7
Evaluate the determinant of each matrix. $$ \left[\begin{array}{cc}{\frac{1}{2}} & {\frac{2}{3}} \\ {\frac{3}{5}} & {\frac{1}{4}}\end{array}\right] $$
View solution Problem 7
Find each sum or difference. $$ \left[\begin{array}{lll}{2} & {1} & {2} \\ {1} & {2} & {1}\end{array}\right]-\left[\begin{array}{lll}{2} & {3} & {2} \\ {3} & {2
View solution Problem 7
Refer to matrices \(A\) and \(B\) at the right. Identify each matrix element. \(A=\left[\begin{array}{rr}{0} & {-1} \\ {1.5} & {3} \\ {7} & {-2}\end{array}\righ
View solution