Problem 33
Question
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(B C\)
Step-by-Step Solution
Verified Answer
Matrix multiplication BC is not possible since \(B\) and \(C\) have incompatible dimensions.
1Step 1: Verify Matrix Dimensions
To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, matrix \(B\) is a 2x2 matrix, and matrix \(C\) is a 3x2 matrix.
2Step 2: Check Compatibility for Multiplication
Matrix \(B\) has 2 columns, and matrix \(C\) has 3 rows. Since the number of columns in \(B\) is not equal to the number of rows in \(C\), matrix multiplication \(BC\) is not possible.
Key Concepts
Matrix MultiplicationMatrix DimensionsNon-conformable Matrices
Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra where two matrices are combined to produce a new matrix. The process involves a systematic method of element pairing and summation. To multiply matrices, each element of a row in the first matrix is multiplied with the corresponding element of a column in the second matrix.
The calculated sums form the resultant matrix. A key point is that matrix multiplication is not commutative, meaning the order of multiplication matters. The product of matrix A and matrix B, denoted as \(A \cdot B\), can differ from the product \(B \cdot A\).
When performing matrix multiplication, it's crucial to follow these steps:
The calculated sums form the resultant matrix. A key point is that matrix multiplication is not commutative, meaning the order of multiplication matters. The product of matrix A and matrix B, denoted as \(A \cdot B\), can differ from the product \(B \cdot A\).
When performing matrix multiplication, it's crucial to follow these steps:
- Identify the elements in the row of the first matrix and elements in the column of the second matrix.
- Multiply corresponding elements from each matrix together and sum these products.
- The resulting value becomes an element in the product matrix.
Matrix Dimensions
Matrix dimensions play a critical role in determining whether matrices can be multiplied together. Each matrix is defined by its dimensions, expressed in terms of rows and columns, denoted as \(m \times n\) where \(m\) is the number of rows, and \(n\) is the number of columns.
Before attempting to multiply matrices, it's necessary to check their dimensions carefully. For two matrices to be eligible for multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. This requirement ensures that each row in the first matrix will pair with a corresponding column in the second matrix during multiplication.
Consider matrices \(B\) and \(C\) for example:
Before attempting to multiply matrices, it's necessary to check their dimensions carefully. For two matrices to be eligible for multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. This requirement ensures that each row in the first matrix will pair with a corresponding column in the second matrix during multiplication.
Consider matrices \(B\) and \(C\) for example:
- Matrix \(B\) is a \(2 \times 2\) matrix, having 2 rows and 2 columns.
- Matrix \(C\) is a \(3 \times 2\) matrix, with 3 rows and 2 columns.
Non-conformable Matrices
Non-conformable matrices refer to matrices that do not meet the criteria needed for certain operations, like multiplication. If matrices do not have compatible dimensions, they are deemed non-conformable. This makes specific mathematical operations impossible to perform. Understanding why this happens is crucial for properly working with matrices.
When matrices are non-conformable, it is usually due to a mismatch between the necessary dimension criteria. As mentioned earlier, for matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Failing to meet this condition results in an inability to carry out multiplication.
In the case of matrices \(B\) and \(C\), as shown in the original step-by-step solution:
When matrices are non-conformable, it is usually due to a mismatch between the necessary dimension criteria. As mentioned earlier, for matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Failing to meet this condition results in an inability to carry out multiplication.
In the case of matrices \(B\) and \(C\), as shown in the original step-by-step solution:
- Matrix \(B\) has 2 columns.
- Matrix \(C\) has 3 rows.
Other exercises in this chapter
Problem 33
For the following exercises, solve the system using the inverse of a \(2 \times 2\) matrix. $$ \begin{array}{l} \frac{8}{5} x-\frac{4}{5} y=\frac{2}{5} \\ -\fra
View solution Problem 33
For the following exercises, solve the system by Gaussian elimination. $$ \left[\begin{array}{lll|l} 1 & 0 & 0 & 31 \\ 0 & 1 & 1 & 45 \\ 0 & 0 & 1 & 87 \end{arr
View solution Problem 33
For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{-2 x^{2}+10 x+4}{(x-1)\l
View solution Problem 33
For the following exercises, use any method to solve the nonlinear system. $$ \begin{aligned} 3 x^{2}-y^{2} &=12 \\ (x-1)^{2}+y^{2} &=1 \end{aligned} $$
View solution