Question

Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.

[ 1 -2 -5 0 4 3 -3 3 0]
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.
A. The matrix is not invertible. In the given matrix the columns do not form a linearly independent set.
B. The matrix is not invertible. the given matrix is A, the equation Ax b has no solution for at least one b in R.
C. The matrix is invertible. The given matrix is not row equivalent to the nx n identity matrix.
D. The matrix is invertible. The given matrix has 3 pivot positions.

Answer 1

This shows 3 pivot position matrixes.

Explanation:

The given matrix is:

`\left[\begin{array}{ccc}1&-2&-5\\0&4&3\\-3&3&0\end{array}\right]`

The option D is correct for this matrix.

The matrix is invertible and the given matrix has 3 pivot positions.

The matrix is invertible if its determinant is nonzero.

Multiply the 3rd row by 1/3.we get:

`\left[\begin{array}{ccc}1&-2&-5\\0&4&3\\-1&1&0\end{array}\right]`

Now, add the first row with third row:

`\left[\begin{array}{ccc}0&-1&-5\\0&4&3\\-1&1&0\end{array}\right]`

Replace third row by first row:

`\left[\begin{array}{ccc}-1&1&0\\0&4&3\\0&-1&-5\end{array}\right]`

This shows 3 pivot position matrixes.

Hence, a matrix is invertible and has 3 pivot positions.