Problem 75
Question
Find \((a)|A|\), (b) \(|B|\), (c) \(A B\), and \((d)\) \(|A B|\). $$ A=\left[\begin{array}{rr} -1 & 0 \\ 0 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 0 \\ 0 & -1 \end{array}\right] $$
Step-by-Step Solution
Verified Answer
(a) The determinant of A, \(|A|\), is -3. (b) The determinant of B, \(|B|\), is -2. (c) The product of A and B, AB, is \( \left[ \begin{array}{c c} -2 & 0 \ 0 & -3 \end{array} \right] \). (d) The determinant of AB, \(|AB|\), is 6.
1Step 1: Calculate the determinant of A
The determinant of A can be calculated by the formula (ad - bc) where a, b, c, and d are the elements of the matrix. In the matrix A, a = -1, b = 0, c = 0 and d = 3. Substituting these values into the formula, the determinant is (-1*3 - 0*0) = -3.
2Step 2: Calculate the determinant of B
Applying the same method as above, the determinant of B is calculated by (ad - bc), where a = 2, b = 0, c = 0, and d = -1. Substituting these values into the formula, the determinant of B is (2*-1 - 0*0) = -2.
3Step 3: Find the product AB
The product of two 2x2 matrices can be calculated by the formula, \( \left[ \begin{array}{c c} ae + bg & af + bh \ ce + dg & cf + dh \end{array} \right] \). Substituting into the formula, AB = \( \left[ \begin{array}{c c} (-1*2 + 0*0) & (-1*0 + 0*-1) \ (0*2 + 3*0) & (0*0 + 3*-1) \end{array} \right] \) = \( \left[ \begin{array}{c c} -2 & 0 \ 0 & -3 \end{array} \right] \)
4Step 4: Find the determinant of AB
Finally, we calculate the determinant of AB by using the formula (ad - bc). Here, a = -2, b = 0, c = 0, d = -3. Substituting these values into the formula, the determinant of AB is (-2*-3 - 0*0) = 6.
Key Concepts
Determinant of a MatrixMatrix Multiplication2x2 MatrixMatrix Algebra
Determinant of a Matrix
Understanding the determinant of a matrix is essential in matrix algebra. It is a scalar value that provides important properties of the matrix. For a 2x2 matrix such as \( A = \left[\begin{array}{rr} -1 & 0 \ 0 & 3 \end{array}\right] \), the determinant can be calculated using the formula \(ad - bc\). Here, \( a, b, c, \) and \( d \) are the elements of the matrix located at positions (1,1), (1,2), (2,1), and (2,2), respectively.
The determinant is significant because it can indicate whether a matrix is invertible, with matrices having zero determinant being non-invertible. Additionally, the determinant can also be interpreted in geometrical terms as the scale factor by which an area or volume is stretched or shrunk by the transformation that the matrix represents.
The determinant is significant because it can indicate whether a matrix is invertible, with matrices having zero determinant being non-invertible. Additionally, the determinant can also be interpreted in geometrical terms as the scale factor by which an area or volume is stretched or shrunk by the transformation that the matrix represents.
Matrix Multiplication
Matrix multiplication is not as straightforward as multiplying individual elements; it involves a specific rule set to combine two matrices. If we have two 2x2 matrices \( A \) and \( B \), the product \( AB \) is found by multiplying rows of \( A \) by columns of \( B \).
The general formula for multiplying two matrices \( A \times B \) where \( A \)=\[\left[\begin{array}{cc} a & b \ c & d \end{array}\right]\] and \( B \) = \[\left[\begin{array}{cc} e & f \ g & h \end{array}\right]\] is given by \[ \left[ \begin{array}{cc} ae + bg & af + bh \ ce + dg & cf + dh \end{array} \right] \]. It's crucial to note that matrix multiplication is not commutative, meaning \( AB \) does not necessarily equal \( BA \) and sometimes \( AB \) and \( BA \) might not both be defined.
The general formula for multiplying two matrices \( A \times B \) where \( A \)=\[\left[\begin{array}{cc} a & b \ c & d \end{array}\right]\] and \( B \) = \[\left[\begin{array}{cc} e & f \ g & h \end{array}\right]\] is given by \[ \left[ \begin{array}{cc} ae + bg & af + bh \ ce + dg & cf + dh \end{array} \right] \]. It's crucial to note that matrix multiplication is not commutative, meaning \( AB \) does not necessarily equal \( BA \) and sometimes \( AB \) and \( BA \) might not both be defined.
2x2 Matrix
A 2x2 matrix is a square array of numbers (or 'elements') that has two rows and two columns. An example of a 2x2 matrix is \( A = \left[\begin{array}{rr} -1 & 0 \ 0 & 3 \end{array}\right] \). The elemental arrangement within a matrix is crucial as it determines the matrix's characteristics and the outcomes of various operations such as addition, subtraction, multiplication, and finding determinants.
In the context of our problem, both \( A \) and \( B \) are 2x2 matrices, which simplifies many operations. For larger matrices, operations can become more complex, but the principles learned from 2x2 matrices can often be extended and applied. Additionally, matrix operations involving 2x2 matrices often serve as an excellent introduction to linear algebra concepts.
In the context of our problem, both \( A \) and \( B \) are 2x2 matrices, which simplifies many operations. For larger matrices, operations can become more complex, but the principles learned from 2x2 matrices can often be extended and applied. Additionally, matrix operations involving 2x2 matrices often serve as an excellent introduction to linear algebra concepts.
Matrix Algebra
Matrix algebra encompasses operations such as addition, subtraction, multiplication, and finding the determinant among others. These operations obey specific rules that may differ from the arithmetic with which we are familiar. Understanding matrix algebra is fundamental for solving linear systems, transforming geometric figures, and dealing with vectors and transformations in higher dimensions.
Through an exercise like the one involving matrices \( A \) and \( B \) and their determinants, we delve into the practical applications of these concepts. Comprehending how to manipulate matrices and interpret the results is a key skill in various fields including computer science, economics, engineering, and more. Developing a solid foundation in matrix algebra is integral for students intending to work with complex systems or pursue advanced studies in mathematics.
Through an exercise like the one involving matrices \( A \) and \( B \) and their determinants, we delve into the practical applications of these concepts. Comprehending how to manipulate matrices and interpret the results is a key skill in various fields including computer science, economics, engineering, and more. Developing a solid foundation in matrix algebra is integral for students intending to work with complex systems or pursue advanced studies in mathematics.
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