For what value of k is the matrix \(\begin{bmatrix} 2\cos 2\theta & 2\cos 2\theta & 6 \\ 1 -2 \sin^2\theta & 2 \cos^2\theta -1 & 3 \\ k & 2k & 1 \end{bmatrix}\) singular?
Understanding Singular Matrices
A matrix is considered singular if its determinant is equal to zero. If the determinant is non-zero, the matrix is called non-singular or invertible.
Simplifying the Matrix Elements
The given matrix is:
\( A = \begin{bmatrix} 2\cos 2\theta & 2\cos 2\theta & 6 \\ 1 -2 \sin^2\theta & 2 \cos^2\theta -1 & 3 \\ k & 2k & 1 \end{bmatrix} \)We can simplify the elements in the second row using standard trigonometric identities:
- \(1 - 2\sin^2\theta = \cos 2\theta\)
- \(2 \cos^2\theta - 1 = \cos 2\theta\)
Substituting these identities, the matrix simplifies to:
\( A = \begin{bmatrix} 2\cos 2\theta & 2\cos 2\theta & 6 \\ \cos 2\theta & \cos 2\theta & 3 \\ k & 2k & 1 \end{bmatrix} \)Analyzing for Linear Dependence in Rows
A powerful property of determinants is that if the rows (or columns) of a matrix are linearly dependent, the determinant is zero, making the matrix singular. Let's look closely at the rows of the simplified matrix:
- Row 1 (\(R_1\)): \([2\cos 2\theta \quad 2\cos 2\theta \quad 6]\)
- Row 2 (\(R_2\)): \([\cos 2\theta \quad \cos 2\theta \quad 3]\)
- Row 3 (\(R_3\)): \([k \quad 2k \quad 1]\)
Let's compare the elements of Row 1 and Row 2:
- First element: \(2\cos 2\theta\) in \(R_1\) and \(\cos 2\theta\) in \(R_2\). \(2\cos 2\theta = 2 \times \cos 2\theta\).
- Second element: \(2\cos 2\theta\) in \(R_1\) and \(\cos 2\theta\) in \(R_2\). \(2\cos 2\theta = 2 \times \cos 2\theta\).
- Third element: \(6\) in \(R_1\) and \(3\) in \(R_2\). \(6 = 2 \times 3\).
We can observe that each element in Row 1 is exactly twice the corresponding element in Row 2. This means that \(R_1\) is a scalar multiple of \(R_2\) (\(R_1 = 2R_2\)).
Determinant and Singularity Condition
Since Row 1 and Row 2 of the matrix are linearly dependent (\(R_1\) is a scalar multiple of \(R_2\)), the determinant of the matrix is guaranteed to be zero. This is a fundamental property of determinants.
\(|A| = 0\)
This condition (\(|A| = 0\)) holds true regardless of the values in the third row, which contains the variable \(k\). The linear dependency between the first two rows alone makes the determinant zero.
Conclusion for the Value of k
Because the determinant of the matrix is always zero, irrespective of the value of \(k\), the matrix is singular for any real value of \(k\).
Revision Table: Key Matrix Properties for Singularity
| Property | Explanation related to Singularity |
|---|---|
| Singular Matrix | A matrix whose determinant is zero. |
| Non-Singular Matrix | A matrix whose determinant is non-zero. It has an inverse. |
| Linear Dependence | If one row (or column) is a linear combination of other rows (or columns), the determinant is zero, and the matrix is singular. |
| Determinant | A scalar value calculated from a square matrix; indicates properties like invertibility and linear dependence. |
Additional Information: Why Linear Dependence Means Determinant is Zero
The determinant of a matrix is closely related to the volume of the parallelepiped spanned by its row or column vectors. If the vectors are linearly dependent, they lie in a lower-dimensional space, and the volume of the parallelepiped they span is zero. Hence, the determinant is zero.
For example, if \(R_1 = c \times R_2\), the row vectors do not span a full 3D space; they are confined to a plane (or a line or a point), which has zero volume in 3D. This is why the determinant is zero.
In systems of linear equations \(Ax = \mathbf{b}\), a singular matrix A implies that there is either no unique solution or infinitely many solutions, depending on \(\mathbf{b}\). For \(Ax = \mathbf{0}\), a singular matrix guarantees non-trivial solutions.
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