Problem 33
Question
Use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(B C\)
Step-by-Step Solution
Verified Answer
Matrix multiplication \( BC \) is not possible because the dimensions are not compatible.
1Step 1: Determine Matrix Dimensions
Matrix \( B \) has dimensions 2x2, which means it has 2 rows and 2 columns. Matrix \( C \) has dimensions 3x2, meaning it has 3 rows and 2 columns. For matrix multiplication to be possible, the number of columns in the first matrix must match the number of rows in the second matrix.
2Step 2: Check Compatibility for Multiplication
To multiply \( B \) and \( C \), \( B \) must have as many columns as \( C \) has rows. \( B \) has 2 columns, but \( C \) has 3 rows. Thus, the number of columns in \( B \) (2) does not equal the number of rows in \( C \) (3).
3Step 3: Conclusion on Matrix Multiplication
Since the columns of \( B \) do not match the rows of \( C \), multiplication \( BC \) is not possible.
Key Concepts
Matrix DimensionsMatrix CompatibilityLinear AlgebraMatrix Operations
Matrix Dimensions
Understanding matrix dimensions is key to performing matrix operations. Each matrix is defined by its dimensions, which are given as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns.
For example, matrix \( B \) is a 2x2 matrix because it has 2 rows and 2 columns. Similarly, matrix \( C \) is a 3x2 matrix as it consists of 3 rows and 2 columns.
Knowing the dimensions helps you understand the structure of the matrix and is essential when it comes to performing operations like multiplication. Without proper dimensions, some operations cannot be performed.
For example, matrix \( B \) is a 2x2 matrix because it has 2 rows and 2 columns. Similarly, matrix \( C \) is a 3x2 matrix as it consists of 3 rows and 2 columns.
Knowing the dimensions helps you understand the structure of the matrix and is essential when it comes to performing operations like multiplication. Without proper dimensions, some operations cannot be performed.
Matrix Compatibility
Matrix compatibility is vital for successful matrix multiplication. Two matrices, say matrix \( A \) and matrix \( D \), can only be multiplied if the number of columns in \( A \) is equal to the number of rows in \( D \).
In the given exercise, matrix \( B \) has 2 columns and matrix \( C \) has 3 rows, making them incompatible for multiplication. Compatibility is crucial because it ensures that each element of a row in the first matrix aligns with a corresponding column in the second matrix, allowing for the dot product to be computed. If this alignment is not possible, the multiplication cannot proceed.
In the given exercise, matrix \( B \) has 2 columns and matrix \( C \) has 3 rows, making them incompatible for multiplication. Compatibility is crucial because it ensures that each element of a row in the first matrix aligns with a corresponding column in the second matrix, allowing for the dot product to be computed. If this alignment is not possible, the multiplication cannot proceed.
Linear Algebra
Linear algebra is a branch of mathematics dealing with vectors, matrices, and linear transformations. It forms the building block for many areas of mathematics and applications such as data science, engineering, and physics.
Understanding matrices and their operations is fundamental in linear algebra. The subject introduces numerous concepts, including vector spaces, matrix transformations, and solving systems of linear equations.
Grasping how matrices operate—starting from simple additions and subtractions to more complex multiplications and inversions—is crucial in linear algebra. It provides the tools needed to manipulate and work with data efficiently.
Understanding matrices and their operations is fundamental in linear algebra. The subject introduces numerous concepts, including vector spaces, matrix transformations, and solving systems of linear equations.
Grasping how matrices operate—starting from simple additions and subtractions to more complex multiplications and inversions—is crucial in linear algebra. It provides the tools needed to manipulate and work with data efficiently.
Matrix Operations
Matrix operations encompass a variety of manipulations you can perform on matrices. Common operations include addition, subtraction, and multiplication.
For matrix multiplication, it's important to ensure that the dimensions align correctly to support the operation. This involves checking compatibility, as demonstrated by the matrices \( B \) and \( C \) in our example.
Other matrix operations include:
For matrix multiplication, it's important to ensure that the dimensions align correctly to support the operation. This involves checking compatibility, as demonstrated by the matrices \( B \) and \( C \) in our example.
Other matrix operations include:
- Addition and Subtraction: Only possible with matrices of the same dimensions, involving element-by-element addition or subtraction.
- Transpose: Flipping a matrix over its diagonal, turning rows into columns and vice versa.
- Determinant and Inverse: Useful in solving linear equations, but only applicable to square matrices.
Other exercises in this chapter
Problem 33
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