Problem 27
Question
Write the given sum as a single-column matrix. $$ \left(\begin{array}{rr} 2 & -3 \\ 1 & 4 \end{array}\right)\left(\begin{array}{r} -2 \\ 5 \end{array}\right)-\left(\begin{array}{rr} -1 & 6 \\ -2 & 3 \end{array}\right)\left(\begin{array}{r} -7 \\ 2 \end{array}\right) $$
Step-by-Step Solution
Verified Answer
The single-column matrix is \( \begin{pmatrix} -38 \\ -2 \end{pmatrix} \).
1Step 1: Multiply the First Matrices
To multiply two matrices, each element of the rows of the first matrix is multiplied with the corresponding element of the columns of the second matrix, and then the products are summed up. For the first product, the first matrix \( \begin{pmatrix} 2 & -3 \ 1 & 4 \end{pmatrix} \) and the second column matrix \( \begin{pmatrix} -2 \ 5 \end{pmatrix} \) are multiplied as follows: Calculate the first row: \[2(-2) + (-3)(5) = -4 - 15 = -19\]Calculate the second row: \[1(-2) + 4(5) = -2 + 20 = 18\]Therefore, the resulting matrix is \( \begin{pmatrix} -19 \ 18 \end{pmatrix} \).
2Step 2: Multiply the Second Matrices
Apply the same multiplication process to the second pair of matrices. For the second product, multiply the matrix \( \begin{pmatrix} -1 & 6 \ -2 & 3 \end{pmatrix} \) with \( \begin{pmatrix} -7 \ 2 \end{pmatrix} \):Calculate the first row:\[(-1)(-7) + 6(2) = 7 + 12 = 19\]Calculate the second row:\[(-2)(-7) + 3(2) = 14 + 6 = 20\]Thus, the resulting matrix is \( \begin{pmatrix} 19 \ 20 \end{pmatrix} \).
3Step 3: Subtract the Matrices
Subtract the second resulting column matrix \( \begin{pmatrix} 19 \ 20 \end{pmatrix} \) from the first resulting column matrix \( \begin{pmatrix} -19 \ 18 \end{pmatrix} \):Subtract the first element:\[-19 - 19 = -38\]Subtract the second element:\[18 - 20 = -2\]The resulting single-column matrix is \( \begin{pmatrix} -38 \ -2 \end{pmatrix} \).
Key Concepts
Matrix MultiplicationMatrix SubtractionColumn MatrixLinear Algebra Operations
Matrix Multiplication
Matrix multiplication is a key concept in linear algebra, allowing us to transform and manipulate data.
When we multiply two matrices, we take a row from the first matrix and multiply it with a column from the second matrix. Each element in the row is multiplied by the corresponding element in the column, and these products are then summed up to give an element in the resulting matrix.
This process is repeated for each row of the first matrix and each column of the second matrix. Ultimately, this creates a new matrix of a size dictated by the original dimensions.
For example, given a matrix of size \( m \times n \) and another of size \( n \times p \), the resulting matrix will have dimensions \( m \times p \).
Keeping track of dimensions is crucial for matrix multiplication to be valid.
When we multiply two matrices, we take a row from the first matrix and multiply it with a column from the second matrix. Each element in the row is multiplied by the corresponding element in the column, and these products are then summed up to give an element in the resulting matrix.
This process is repeated for each row of the first matrix and each column of the second matrix. Ultimately, this creates a new matrix of a size dictated by the original dimensions.
For example, given a matrix of size \( m \times n \) and another of size \( n \times p \), the resulting matrix will have dimensions \( m \times p \).
Keeping track of dimensions is crucial for matrix multiplication to be valid.
Matrix Subtraction
Matrix subtraction is an operation where each element of one matrix is subtracted from the corresponding element in another matrix.
This requires both matrices to be of the same dimension, meaning they must have the same number of rows and columns.
If the matrices are not of the same size, you can't perform matrix subtraction.
Imagine subtracting one number from another across two parallel lists. That's essentially what matrix subtraction does, but for an entire table of numbers.
In terms of application, matrix subtraction can show differences or changes in data sets, often used together with addition and multiplication in linear algebra problems.
This requires both matrices to be of the same dimension, meaning they must have the same number of rows and columns.
If the matrices are not of the same size, you can't perform matrix subtraction.
Imagine subtracting one number from another across two parallel lists. That's essentially what matrix subtraction does, but for an entire table of numbers.
In terms of application, matrix subtraction can show differences or changes in data sets, often used together with addition and multiplication in linear algebra problems.
Column Matrix
A column matrix, also known as a column vector, is simply a matrix that consists of only one column.
These types of matrices play a significant role in various calculations and representations in linear algebra.
A column matrix may look like \( \begin{pmatrix} a \ b \ c \end{pmatrix} \), where \( a \), \( b \), and \( c \) are elements that can represent coordinates or vector components in 3D space, for instance.
Even though they're simple by design, column matrices are crucial in intermediate steps of matrix operations, like multiplication.
When multiplying another matrix by a column vector, we often end up with another column matrix, as seen in many practical applications, from physics to economics.
These types of matrices play a significant role in various calculations and representations in linear algebra.
A column matrix may look like \( \begin{pmatrix} a \ b \ c \end{pmatrix} \), where \( a \), \( b \), and \( c \) are elements that can represent coordinates or vector components in 3D space, for instance.
Even though they're simple by design, column matrices are crucial in intermediate steps of matrix operations, like multiplication.
When multiplying another matrix by a column vector, we often end up with another column matrix, as seen in many practical applications, from physics to economics.
Linear Algebra Operations
Linear algebra operations include a variety of methods used to solve problems involving linear equations and linear transformations.
Some of the most common operations are addition, subtraction, and multiplication of matrices.
Beyond these, finding the determinant, inverse, and transpose of matrices are also pivotal techniques.
In a practical sense, these operations help in solving systems of linear equations, performing transformations, and managing data.
Mastering these operations requires practice but provides essential tools to handle more complex mathematical modeling and data analysis tasks.
Some of the most common operations are addition, subtraction, and multiplication of matrices.
Beyond these, finding the determinant, inverse, and transpose of matrices are also pivotal techniques.
In a practical sense, these operations help in solving systems of linear equations, performing transformations, and managing data.
Mastering these operations requires practice but provides essential tools to handle more complex mathematical modeling and data analysis tasks.
- Matrix Multiplication
- Matrix Addition and Subtraction
- Inversion and Determinants
Other exercises in this chapter
Problem 27
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Without expanding, show that \(\left|\begin{array}{ccc}1 & 1 & 1 \\\ \frac{1}{a} & \frac{1}{b} & \frac{1}{c} \\ b c & a c & a b\end{array}\right|=0\)
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