A matrix is not invertible (or singular) if and only if its determinant is zero. So, we need to find the values of λ that make the determinant of the matrix zero.

The matrix is:

1 λ 0
3 2 0
1 2 1

The determinant of a 3x3 matrix

a b c
d e f
g h i

is given by the formula:

a(ei−fh)−b(di−fg)+c(dh−eg)

So, the determinant of our matrix is:

1*(2*1 - 0*2) - λ*(3*1 - 0*1) + 0*(3*2 - 1*1)

Simplify this to:

2 - 3λ

Set the determinant equal to zero and solve for λ:

2 - 3λ = 0
3λ = 2
λ = 2/3

So, the matrix is not invertible for λ = 2/3.

Question

A matrix is not invertible (or singular) if and only if its determinant is zero. So, we need to find the values of λ that make the determinant of the matrix zero.

The matrix is:

1 λ 0
3 2 0
1 2 1

The determinant of a 3x3 matrix

a b c
d e f
g h i

is given by the formula:

a(ei−fh)−b(di−fg)+c(dh−eg)

So, the determinant of our matrix is:

1*(2*1 - 0*2) - λ*(3*1 - 0*1) + 0*(3*2 - 1*1)

Simplify this to:

2 - 3λ

Set the determinant equal to zero and solve for λ:

2 - 3λ = 0
3λ = 2
λ = 2/3

So, the matrix is not invertible for λ = 2/3.

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