Let \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {\rm x + y}& \rm y\\ {\rm 2x}&{\rm x - y} \end{array}} \right],\;\rm B = \left[ {\begin{array}{*{20}{c}} 2\\ { - 1} \end{array}} \right]\) and \(\rm C = \left[ {\begin{array}{*{20}{c}} 3\\ 2 \end{array}} \right]\) . If AB = C, then what is the value of the determinant of the matrix A?
-14
Understanding the Matrix Problem
The question asks us to find the determinant of matrix A, given the matrices A, B, and C, and the equation AB = C. To solve this, we first need to use the matrix equation AB = C to find the values of the variables x and y present in matrix A. Once we have x and y, we can determine the specific form of matrix A and then calculate its determinant.
Given Matrices
We are given the following matrices:
- Matrix A: \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {\rm x + y}& \rm y\\ {\rm 2x}&{\rm x - y} \end{array}} \right]\)
- Matrix B: \(\rm B = \left[ {\begin{array}{*{20}{c}} 2\\ { - 1} \end{array}} \right]\)
- Matrix C: \(\rm C = \left[ {\begin{array}{*{20}{c}} 3\\ 2 \end{array}} \right]\)
We are also given the matrix equation \(\rm AB = C\).
Performing Matrix Multiplication AB
Let's perform the matrix multiplication of A and B:
\(\rm AB = \left[ {\begin{array}{*{20}{c}} {\rm x + y}& \rm y\\ {\rm 2x}&{\rm x - y} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} 2\\ { - 1} \end{array}} \right]\)
To multiply these matrices, we multiply the rows of the first matrix by the columns of the second matrix.
- The element in the first row of AB is: \((x+y) \times 2 + (y) \times (-1) = 2x + 2y - y = 2x + y\)
- The element in the second row of AB is: \((2x) \times 2 + (x-y) \times (-1) = 4x - (x - y) = 4x - x + y = 3x + y\)
So, the product matrix AB is:
\(\rm AB = \left[ {\begin{array}{*{20}{c}} 2x + y\\ 3x + y \end{array}} \right]\)
Equating AB to C and Solving for x and y
We are given that AB = C. Equating the elements of the resulting matrix AB to the elements of matrix C:
\(\left[ {\begin{array}{*{20}{c}} 2x + y\\ 3x + y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 3\\ 2 \end{array}} \right]\)
This gives us a system of two linear equations:
- \(2x + y = 3\)
- \(3x + y = 2\)
We can solve this system using the elimination method. Subtract equation (1) from equation (2):
\((3x + y) - (2x + y) = 2 - 3\)
\(3x + y - 2x - y = -1\)
\(x = -1\)
Now substitute the value of x = -1 into equation (1):
\(2(-1) + y = 3\)
\(-2 + y = 3\)
\(y = 3 + 2\)
\(y = 5\)
So, we have found the values x = -1 and y = 5.
Determining Matrix A
Now that we have the values of x and y, we can substitute them back into the expression for matrix A:
\({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {\rm x + y}& \rm y\\ {\rm 2x}&{\rm x - y} \end{array}} \right]\)
\({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {-1 + 5}& 5\\ {2(-1)}&{-1 - 5} \end{array}} \right]\)
\({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 4& 5\\ {-2}&{-6} \end{array}} \right]\)
Calculating the Determinant of Matrix A
Matrix A is a 2x2 matrix. The determinant of a 2x2 matrix \(\left[ {\begin{array}{*{20}{c}} a& b\\ c& d \end{array}} \right]\) is calculated as \(ad - bc\).
For matrix \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 4& 5\\ {-2}&{-6} \end{array}} \right]\), we have a = 4, b = 5, c = -2, and d = -6.
Determinant of A = \((4) \times (-6) - (5) \times (-2)\)
Determinant of A = \(-24 - (-10)\)
Determinant of A = \(-24 + 10\)
Determinant of A = \(-14\)
Conclusion
The value of the determinant of matrix A is -14.
Revision Table: Key Concepts
| Concept | Description | Formula/Method |
|---|---|---|
| Matrix Multiplication (AB) | Multiplying matrices A and B results in a new matrix where each element is the sum of products of corresponding elements from a row of A and a column of B. | For \({\rm{A}}_{m \times n}\) and \({\rm{B}}_{n \times p}\), \((AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}\) |
| System of Linear Equations | A set of equations with the same variables that can be solved simultaneously to find the values of the variables. | Methods include substitution, elimination, matrix methods (like Cramer's rule, inverse matrix). |
| Determinant of a 2x2 Matrix | A scalar value computed from the elements of a square matrix. For a 2x2 matrix \(\left[ {\begin{array}{*{20}{c}} a& b\\ c& d \end{array}} \right]\), it's \(ad - bc\). | \(\det(A) = ad - bc\) |
Additional Information: Properties and Applications
- Determinant Properties: The determinant has many useful properties. For example, \(\det(AB) = \det(A) \det(B)\), which could be an alternative (though perhaps more complex in this case) way to approach problems involving matrix products if determinants of B and C were calculable directly. Also, a matrix is invertible if and only if its determinant is non-zero.
- Matrix Equations: Equations like AB = C are fundamental in linear algebra and represent systems of linear equations in a compact form. Solving for an unknown matrix or unknown variables within a matrix often involves matrix operations or converting back to linear systems.
- Applications of Determinants: Determinants are used to find the inverse of a matrix, solve systems of linear equations using Cramer's rule, calculate the area of a triangle or volume of a tetrahedron defined by vectors, and in transformations (e.g., how area/volume scales under a linear transformation).
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