If `[{:(1,-3,2),(2,-8,5),(4,2,lambda):}]` is not an invertible matrix, then what is the value of `lambda` ?

Options

A. -1
B. 0
C. 1
D. 2

Explanation

To find the value of \( \lambda \) such that the matrix

\[ A = \begin{pmatrix} 1 & -3 & 2 \\ 2 & -8 & 5 \\ 4 & 2 & \lambda \end{pmatrix} \]

is not invertible, we need to set the determinant of the matrix \( A \) to zero. A matrix is not invertible (or singular) if its determinant is zero.

### Step 1: Calculate the Determinant

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

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

is given by the formula:

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

For our matrix \( A \):

- \( a = 1, b = -3, c = 2 \)
- \( d = 2, e = -8, f = 5 \)
- \( g = 4, h = 2, i = \lambda \)

Substituting these values into the determinant formula, we have:

\[ \text{det}(A) = 1((-8)\lambda - (5)(2)) - (-3)((2)(\lambda) - (5)(4)) + 2((2)(2) - (-8)(4)) \]

### Step 2: Simplify Each Term

Now, we simplify each term:

1. **First term**:
\[ 1((-8)\lambda - 10) = -8\lambda - 10 \]

2. **Second term**:
\[ -(-3)((2\lambda) - 20) = 3(2\lambda - 20) = 6\lambda - 60 \]

3. **Third term**:
\[ 2((2)(2) - (-8)(4)) = 2(4 + 32) = 2(36) = 72 \]

### Step 3: Combine All Terms

Now, we combine all the terms we calculated:

\[ -8\lambda - 10 + 6\lambda - 60 + 72 = 0 \]

Combining like terms gives:

\[ (-8\lambda + 6\lambda) + (-10 - 60 + 72) = 0 \]

This simplifies to:

\[ -2\lambda + 2 = 0 \]

### Step 4: Solve for \( \lambda \)

Now, we solve for \( \lambda \):

\[ -2\lambda + 2 = 0 \implies -2\lambda = -2 \implies \lambda = 1 \]

Thus, the value of \( \lambda \) for which the matrix is not invertible is

\[ \boxed{1} \]