If matrix [0 1 -2   -1 0 3   k -3 0] is singular then value of k 

For the matrix 

\[
\begin{bmatrix}
0 & 1 & -2 \\
-1 & 0 & 3 \\
k & -3 & 0
\end{bmatrix}
\]

to be singular, its determinant must be zero.

The determinant of a \(3 \times 3\) matrix 

\[
\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{bmatrix}
\]

is given by:

\[
\text{det} = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

Applying this to our matrix:

\[
a = 0, \; b = 1, \; c = -2, \; d = -1, \; e = 0, \; f = 3, \; g = k, \; h = -3, \; i = 0
\]

So,

\[
\text{det} = 0 \cdot (0 \cdot 0 - 3 \cdot (-3)) - 1 \cdot (-1 \cdot 0 - 3 \cdot k) + (-2) \cdot (-1 \cdot (-3) - 0 \cdot k)
\]

Simplify each term:

1. The first term is \(0 \cdot (0 + 9) = 0\).
2. The second term is \(-1 \cdot (0 - 3k) = 3k\).
3. The third term is \(-2 \cdot (3) = -6\).

So,

\[
\text{det} = 0 + 3k - 6
\]

For the matrix to be singular, we set the determinant to zero:

\[
3k - 6 = 0
\]

Solving for \(k\):

\[
3k = 6 \implies k = 2
\]

Thus, the value of \(k\) that makes the matrix singular is \( \boxed{2} \).

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How do determinants relate to the invertibility of matrices?
2. What does it mean for a matrix to be singular in terms of its rank?
3. How does changing other elements in the matrix affect the value of \(k\) for singularity?
4. Can singular matrices be used in systems of linear equations?
5. What is the significance of the zero determinant in geometry?

**Tip:** For a \(3 \times 3\) matrix, checking if it’s singular is often easiest by calculating its determinant directly.

If matrix [0 1 -2   -1 0 3   k -3 0] is singular then value of k

Learn how to find the value of k that makes the matrix [0 1 -2; -1 0 3; k -3 0] singular. The solution involves calculating the determinant of a 3x3 matrix and setting it equal to zero to determine when the matrix is non-invertible. Suitable for Grades 11-12.

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