Problem 14
Question
Use the matrices below to perform scalar multiplication. \(A=\left[\begin{array}{rr}4 & 6 \\ 13 & 12\end{array}\right], B=\left[\begin{array}{rr}3 & 9 \\ 21 & 12 \\ 0 & 64\end{array}\right], C=\left[\begin{array}{rrrr}16 & 3 & 7 & 18 \\ 90 & 5 & 3 & 29\end{array}\right], D=\left[\begin{array}{rrr}18 & 12 & 13 \\ 8 & 14 & 6 \\\ 7 & 4 & 21\end{array}\right]\) \(-2 B\)
Step-by-Step Solution
Verified Answer
The resulting matrix after scalar multiplication by \(-2\) is \(
\left[\begin{array}{rr}
-6 & -18 \\
-42 & -24 \\
0 & -128
\end{array}\right]
\).
1Step 1: Identify the Matrix
We need to perform scalar multiplication on matrix \(B\), which is given as:\[B=\left[\begin{array}{rr}3 & 9 \ 21 & 12 \ 0 & 64\end{array}\right]\]
2Step 2: Understand Scalar Multiplication
Scalar multiplication involves multiplying each element of the matrix by a scalar. Here, the scalar is \(-2\). We'll apply this to each entry of matrix \(B\).
3Step 3: Multiply Each Element by Scalar
Perform the multiplication for each element in matrix \(B\).- First row: \(-2 \cdot 3 = -6, \quad -2 \cdot 9 = -18\)- Second row: \(-2 \cdot 21 = -42, \quad -2 \cdot 12 = -24\)- Third row: \(-2 \cdot 0 = 0, \quad -2 \cdot 64 = -128\)
4Step 4: Formulate the Resulting Matrix
Combine the results of the multiplications into a new matrix, as follows:\[-2B = \left[\begin{array}{rr}-6 & -18 \-42 & -24 \0 & -128\end{array}\right]\]
Key Concepts
Matrix MultiplicationStep-by-step SolutionAlgebraic Operations
Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra that involves combining two matrices to produce a new matrix. A matrix is an array of numbers arranged in rows and columns. Unlike scalar multiplication, which involves a single number, matrix multiplication considers the structure and alignment of two matrices.
To multiply matrices, you must adhere to specific rules. First, check if the two matrices can be multiplied by examining their dimensions. Matrix multiplication is defined when the number of columns in the first matrix matches the number of rows in the second matrix. Once confirmed, proceed by multiplying the entries of the rows of the first matrix with the corresponding entries of the columns of the second matrix, then summing the products.
It's important to recognize that matrix multiplication is not commutative, meaning the order in which you multiply matters. Different orders can yield different results, or the multiplication may not be defined at all.
To multiply matrices, you must adhere to specific rules. First, check if the two matrices can be multiplied by examining their dimensions. Matrix multiplication is defined when the number of columns in the first matrix matches the number of rows in the second matrix. Once confirmed, proceed by multiplying the entries of the rows of the first matrix with the corresponding entries of the columns of the second matrix, then summing the products.
It's important to recognize that matrix multiplication is not commutative, meaning the order in which you multiply matters. Different orders can yield different results, or the multiplication may not be defined at all.
Step-by-step Solution
Understanding a detailed step-by-step solution can significantly enhance one’s comprehension of scalar multiplication. When dealing with matrices, as with matrix 'B' in the exercise, approach the problem methodically. Start by clearly identifying the matrix on which to perform operations.
Once the matrix is identified, focus on understanding the operation needed. Scalar multiplication, in this context, requires multiplying each element of matrix 'B' by a specific number, which is \(-2\) in our example. By breaking down the task into manageable steps, such as multiplying each row independently and recalculating row by row, the overall process becomes clearer.
Once the matrix is identified, focus on understanding the operation needed. Scalar multiplication, in this context, requires multiplying each element of matrix 'B' by a specific number, which is \(-2\) in our example. By breaking down the task into manageable steps, such as multiplying each row independently and recalculating row by row, the overall process becomes clearer.
Algebraic Operations
Algebraic operations on matrices, including scalar multiplication, are essential in various scientific and engineering applications. Scalar multiplication is straightforward: you take each entry in a matrix and multiply it by a scalar value. This operation maintains the matrix's structure while changing its magnitude or direction, depending on the scalar's value.
Beyond scalar multiplication, matrix algebra includes more complex operations such as addition, subtraction, and multiplication of two matrices. Each of these operations follows specific rules and serves a different purpose within mathematical and practical applications.
Understanding these basic operations is crucial as they form the foundation for more advanced studies in linear transformations and systems of equations. Rooting your learning in a strong grasp of basic algebraic operations facilitates greater ease with complex matrix calculations.
Beyond scalar multiplication, matrix algebra includes more complex operations such as addition, subtraction, and multiplication of two matrices. Each of these operations follows specific rules and serves a different purpose within mathematical and practical applications.
Understanding these basic operations is crucial as they form the foundation for more advanced studies in linear transformations and systems of equations. Rooting your learning in a strong grasp of basic algebraic operations facilitates greater ease with complex matrix calculations.
Other exercises in this chapter
Problem 14
For the following exercises, find the multiplicative inverse of each matrix, if it exists. $$\left[\begin{array}{rr}-2 & 2 \\ 3 & 1\end{array}\right]$$
View solution Problem 14
Solve the system of nonlinear equations using elimination. $$ \begin{array}{r} y^{2}-x^{2}=9 \\ 3 x^{2}+2 y^{2}=8 \end{array} $$
View solution Problem 14
For the following exercises, use the matrices below to perform scalar multiplication. $$ A=\left[\begin{array}{cc}{4} & {6} \\ {13} & {12}\end{array}\right], B=
View solution Problem 14
Solve each system by substitution. $$ \begin{aligned} 4 x-3 y+5 z &=31 \\ -x+2 y+4 z &=20 \\ x+5 y-2 z &=-29 \end{aligned} $$
View solution