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).
Similar Questions
If \({\rm{E}}\left( {\rm{\theta }} \right) = \left[ {\begin{array}{*{20}{c}} {\cos {\rm{\theta }}}&{\sin {\rm{\theta }}}\\ { - \sin {\rm{\theta \;}}}&{\cos {\rm{\theta }}} \end{array}} \right]\) then E(α) E(β) is equal to
Consider the following in respect of matrices A, B and C of same order:
1) (A + B + C)' = A' + B’ + C’
2) (AB)’ = A’B’
3) (ABC)’ = C’B’A’
Where A’ is the transpose of the matrix A.
Which of the above are correct?If \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 1&0&{ - 2}\\ 2&{ - 3}&4 \end{array}} \right]\) , then the matrix X for which 2X + 3A = 0 holds true is
Let \(A = \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}\) and (mI + nA) 2= A where m, n are positive real numbers and I is the identify matrix. What is (m + n) equal to?
If \(A = \left[ {\begin{array}{c} 1&{ - 1}\\ { - 1}&1 \end{array}} \right],\) then the expression A 3- 2A 2is
If \(A = \left( {\begin{array}{*{20}{c}} 1&2\\ 2&3\\ 3&4 \end{array}} \right)\) and \(B = \left( {\begin{array}{*{20}{c}} 1&2\\ 2&1 \end{array}} \right),\) then which one of the following is correct?
A square matrix A is called orthogonal if_______ where A’ is the transpose of A.
Consider the following in respect of matrices A and B of same order:
1) A 2– B 2= (A + B) (A – B)
2) (A – I) (I + A) = O ⇔ A 2= I
Where I is the identity matrix and O is the null matrix.
Which of the above is/are correct?If A is a 2 × 3 matrix and AB is a 2 × 5 matrix, then B must be a
If \(A = \left( {\begin{array}{} 1&2\\ 2&3 \end{array}} \right)\) and A 2– kA – I 2= 0, where I 2is the 2 × 2 identity matrix, then what is the value of k?
Important Questions from Operations on Matrices
The adjoint of matrix \(\left[ {\begin{array}{*{20}{c}} a&b\\ c&d \end{array}} \right]\)is
If \({\rm{E}}\left( {\rm{\theta }} \right) = \left[ {\begin{array}{*{20}{c}} {\cos {\rm{\theta }}}&{\sin {\rm{\theta }}}\\ { - \sin {\rm{\theta \;}}}&{\cos {\rm{\theta }}} \end{array}} \right]\) then E(α) E(β) is equal to
Consider the following in respect of matrices A, B and C of same order:
1) (A + B + C)' = A' + B’ + C’
2) (AB)’ = A’B’
3) (ABC)’ = C’B’A’
Where A’ is the transpose of the matrix A.
Which of the above are correct?If \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 1&0&{ - 2}\\ 2&{ - 3}&4 \end{array}} \right]\) , then the matrix X for which 2X + 3A = 0 holds true is
Find a matrix X such that 2A + B + X = 0 , where
\(A=\begin{bmatrix} -1 & 2 \\\ 3 & 4 \end{bmatrix} \ \text{and} \;\rm B =\ \begin{bmatrix} 3 & -2 \\\ 1 & 5 \end{bmatrix} \ ?\)
