This problem involves finding the values of variables \(x\) and \(y\) within a 3x3 matrix whose determinant is given as a complex number. To solve this, we need to calculate the determinant of the given matrix, which contains complex number entries, and then equate the real and imaginary parts of the resulting complex number with the real and imaginary parts of the given determinant value.
The given matrix is:
Column 1
Column 2
Column 3
Row 1
\(x\)
\(-3i\)
\(1\)
Row 2
\(y\)
\(1\)
\(i\)
Row 3
\(0\)
\(2i\)
\(-i\)
We are given that the determinant of this matrix is equal to \(6+11i\).
Let's calculate the determinant of the matrix. We can expand along the first row:
\(\det \left( {\begin{array}{*{20}{c}} x&-3i&1\\ y&1&{i}\\ 0&2i&-i \end{array}} \right) = x \cdot \begin{vmatrix} 1 & i \\ 2i & -i \end{vmatrix} - (-3i) \cdot \begin{vmatrix} y & i \\ 0 & -i \end{vmatrix} + 1 \cdot \begin{vmatrix} y & 1 \\ 0 & 2i \end{vmatrix}\)
\(\det(A) = 2x - xi - 3(-1)y + 2yi\)\(\det(A) = 2x - xi + 3y + 2yi\)
Now, group the real and imaginary parts of the determinant:
\(\det(A) = (2x + 3y) + (-x + 2y)i\)
We are given that the determinant is \(6 + 11i\). Therefore, we can equate the real and imaginary parts:
Equating Real Parts:
\(2x + 3y = 6\) (Equation 1)
Equating Imaginary Parts:
\(-x + 2y = 11\) (Equation 2)
Now we have a system of two linear equations with two variables, \(x\) and \(y\). We can solve this system using substitution or elimination.
Let's use the substitution method. From Equation 2, we can express \(x\) in terms of \(y\):
\(-x = 11 - 2y\)
\(x = -11 + 2y\) (Equation 3)
Now substitute Equation 3 into Equation 1:
\(2(-11 + 2y) + 3y = 6\)
Expand and simplify:
\(-22 + 4y + 3y = 6\)
\(-22 + 7y = 6\)
Add 22 to both sides:
\(7y = 6 + 22\)
\(7y = 28\)
Divide by 7:
\(y = \frac{28}{7}\)
\(y = 4\)
Now substitute the value of \(y = 4\) back into Equation 3 to find \(x\):
\(x = -11 + 2(4)\)
\(x = -11 + 8\)
\(x = -3\)
Thus, the values of \(x\) and \(y\) are \(-3\) and \(4\), respectively.
The final answer is \(x = -3\) and \(y = 4\).
Revision Table: Complex Matrix Determinant
Concept
Description
Application in this problem
Determinant of a 3x3 Matrix
A scalar value calculated from the elements of a square matrix.
Calculated the determinant of the given 3x3 matrix with complex entries.
Complex Numbers
Numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit (\(i^2 = -1\)).
Matrix elements and the determinant value were complex numbers.
Equating Complex Numbers
Two complex numbers \(a + bi\) and \(c + di\) are equal if and only if their real parts are equal (\(a=c\)) and their imaginary parts are equal (\(b=d\)).
Used to form a system of linear equations by equating the real and imaginary parts of the calculated determinant and the given determinant value.
System of Linear Equations
A set of two or more linear equations involving the same variables.
Formed a system of two equations (\(2x+3y=6\) and \(-x+2y=11\)) to solve for \(x\) and \(y\).
Additional Information: Determinants and Complex Numbers
Determinants are fundamental in linear algebra and have various applications, such as finding the inverse of a matrix, solving systems of linear equations (using Cramer's rule), and understanding geometric transformations. When matrices contain complex numbers, the properties of determinants still hold, but the calculations involve complex arithmetic.
Calculating the determinant of a matrix with complex entries follows the same rules as for matrices with real entries. The key is to perform complex number arithmetic correctly at each step. This includes adding, subtracting, multiplying, and dividing complex numbers as needed.
In this problem, the determinant calculation resulted in a complex number expression \((2x + 3y) + (-x + 2y)i\). Since this expression must equal the given complex number \(6 + 11i\), we applied the property that two complex numbers are equal if and only if their corresponding real and imaginary parts are equal. This technique is commonly used when solving equations involving complex numbers.
Solving the resulting system of linear equations is a standard algebraic procedure. Methods like substitution, elimination, or matrix methods (like Cramer's rule or using inverse matrices) can be used. In this case, the substitution method was a straightforward way to find the values of \(x\) and \(y\).
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