Problem 11
Question
Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{ll} {1} & {3} \\ {3} & {4} \\ {5} & {6} \end{array}\right], \quad B=\left[\begin{array}{rr} {2} & {-1} \\ {3} & {-2} \\ {0} & {1} \end{array}\right] $$
Step-by-Step Solution
Verified Answer
The results are: a. \(A+B=\left[\begin{array}{ll} {3} & {2} \ {6} & {2} \ {5} & {7} \end{array}\right]\)b. \(A-B=\left[\begin{array}{ll} {-1} & {4} \ {0} & {6} \ {5} & {5} \end{array}\right]\) c. \(-4A=\left[\begin{array}{ll} {-4} & {-12} \ {-12} & {-16} \ {-20} & {-24} \end{array}\right]\) d. \(3A+2B=\left[\begin{array}{ll} {5} & {7} \ {12} & {8} \ {15} & {20} \end{array}\right]\)
1Step 1: Matrix Addition
Add the corresponding elements of matrix A and B to calculate A+B. This will form a new matrix where each element is the sum of the corresponding elements in matrices A and B.
2Step 2: Matrix Subtraction
Subtract the elements of matrix B from those of matrix A to find A-B. The resulting matrix will be formed by subtracting each of the corresponding elements in matrices A and B.
3Step 3: Scalar Multiplication with a Matrix
Multiply each element in matrix A by the scalar -4. The new matrix, -4A, will have each of its elements four times the corresponding element in A, with opposite signs.
4Step 4: Combination of Addition and Scalar Multiplication
Multiply elements in matrix A by 3 and in matrix B by 2, and then add the resulting matrices. This new combined operation matrix, 3A+2B, will have each of its elements obtained by multiplying the respective elements in A and B by 3 and 2, and then adding these together.
Key Concepts
Matrix AdditionMatrix SubtractionScalar Multiplication with a MatrixMatrix Algebra
Matrix Addition
Understanding how to perform matrix addition is fundamental in matrix algebra. It involves adding the corresponding elements of two matrices together. The key here is that the matrices must be of the same size, meaning they have the same number of rows and columns.
For example, when we have two matrices, A and B, the sum of these matrices, denoted as \(A + B\), is computed by adding the elements that are in the same position in each matrix. Here’s how it would look for a 2x2 matrix:
\begin{align*}A &= \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix},B &= \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix},\(A + B\) &= \begin{bmatrix}a_{11}+b_{11} & a_{12}+b_{12} \a_{21}+b_{21} & a_{22}+b_{22} \end{bmatrix}.\rend{align*}
Each entry in the resulting matrix is the sum of the corresponding entries in matrices A and B. Fundamentally, it's like adding numbers, but instead, you're adding arrays of numbers in an organized way.
For example, when we have two matrices, A and B, the sum of these matrices, denoted as \(A + B\), is computed by adding the elements that are in the same position in each matrix. Here’s how it would look for a 2x2 matrix:
\begin{align*}A &= \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix},B &= \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix},\(A + B\) &= \begin{bmatrix}a_{11}+b_{11} & a_{12}+b_{12} \a_{21}+b_{21} & a_{22}+b_{22} \end{bmatrix}.\rend{align*}
Each entry in the resulting matrix is the sum of the corresponding entries in matrices A and B. Fundamentally, it's like adding numbers, but instead, you're adding arrays of numbers in an organized way.
Matrix Subtraction
Matrix subtraction is also a straightforward concept if you've grasped matrix addition. It operates under the same premise: only matrices of the same dimensions can be subtracted from each other.
To find the difference of two matrices, A and B, denoted as \(A - B\), we subtract the elements of B from the corresponding elements of A. The process is similar to addition, but instead of adding, we subtract:
\begin{align*}A &= \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix},B &= \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix},\(A - B\) &= \begin{bmatrix}a_{11}-b_{11} & a_{12}-b_{12} \a_{21}-b_{21} & a_{22}-b_{22} \end{bmatrix}.\rend{align*}
Each entry in the result is the difference between the matching entries of A and B. Just remember, subtraction has direction, which means \(A - B\) is generally not the same as \(B - A\).
To find the difference of two matrices, A and B, denoted as \(A - B\), we subtract the elements of B from the corresponding elements of A. The process is similar to addition, but instead of adding, we subtract:
\begin{align*}A &= \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix},B &= \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix},\(A - B\) &= \begin{bmatrix}a_{11}-b_{11} & a_{12}-b_{12} \a_{21}-b_{21} & a_{22}-b_{22} \end{bmatrix}.\rend{align*}
Each entry in the result is the difference between the matching entries of A and B. Just remember, subtraction has direction, which means \(A - B\) is generally not the same as \(B - A\).
Scalar Multiplication with a Matrix
Another important operation in matrix algebra is scalar multiplication. This involves taking a matrix and multiplying every element within it by a single number, which is known as a 'scalar'.
Let's consider a scalar \(c\) and a matrix A. The product of the scalar and the matrix, denoted as \(cA\), is computed by multiplying each entry \(a_{ij}\) of A by \(c\):
\begin{align*}A &= \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix},cA &= \begin{bmatrix} ca_{11} & ca_{12} \ ca_{21} & ca_{22} \end{bmatrix}.\rend{align*}
Through scalar multiplication, the size of the matrix remains the same, but each element is scaled by the factor \(c\). This operation is useful when we want to change the scale of the matrix entries or simply include matrix terms in a larger algebraic expression.
Let's consider a scalar \(c\) and a matrix A. The product of the scalar and the matrix, denoted as \(cA\), is computed by multiplying each entry \(a_{ij}\) of A by \(c\):
\begin{align*}A &= \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix},cA &= \begin{bmatrix} ca_{11} & ca_{12} \ ca_{21} & ca_{22} \end{bmatrix}.\rend{align*}
Through scalar multiplication, the size of the matrix remains the same, but each element is scaled by the factor \(c\). This operation is useful when we want to change the scale of the matrix entries or simply include matrix terms in a larger algebraic expression.
Matrix Algebra
Matrix algebra encompasses operations like addition, subtraction, and scalar multiplication, as well as more complex operations such as matrix multiplication and inversion. It is important to note that these operations have unique rules that differ from regular arithmetic.
For example, in matrix algebra, the commutative property of addition \(A + B = B + A\) holds true, but this is not always the case for matrix multiplication. Additionally, multiplying matrices is not a simple element-by-element process—it involves a pattern of multiplication and addition known as the 'dot product'.
Learning matrix algebra is critical for various applications in mathematics, physics, engineering, computer science, economics, and more. It provides a framework for solving systems of linear equations, transforming geometric objects, and working with vectors and vector spaces. Mastery of matrix operations thus lays the groundwork for advancing in these fields.
For example, in matrix algebra, the commutative property of addition \(A + B = B + A\) holds true, but this is not always the case for matrix multiplication. Additionally, multiplying matrices is not a simple element-by-element process—it involves a pattern of multiplication and addition known as the 'dot product'.
Learning matrix algebra is critical for various applications in mathematics, physics, engineering, computer science, economics, and more. It provides a framework for solving systems of linear equations, transforming geometric objects, and working with vectors and vector spaces. Mastery of matrix operations thus lays the groundwork for advancing in these fields.
Other exercises in this chapter
Problem 10
Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{lll} {0} & {2} & {0} \\ {3} & {3} & {2}
View solution Problem 10
Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\\{\begin{aligned} 2 w+x-2 y-z &=3 \\ w-2
View solution Problem 11
Write the system of linear equations represented by the augmented matrix. Use \(x, y,\) and \(z,\) or, if necessary, \(w, x, y\) and \(z,\) for the variables. $
View solution Problem 11
Use Cramer’s Rule to solve each system. $$ \left\\{\begin{array}{l} {x+y=7} \\ {x-y=3} \end{array}\right. $$
View solution