Question

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

0	2	−4
1	0	2
−2	−6	8

Choose the correct answer below.

A.

The matrix is invertible. The columns of the given matrix span are linearly dependent.

B.

The matrix is not invertible. If the given matrix is​ A, the equation

Ax=b has a solution for at least one b in set of real numbers ℝ3.

C.The matrix is invertible. The given matrix has 3 pivot positions.

D.The matrix is not invertible. If the given matrix is​ A, A is not row equivalent to the n times n×n identity matrix.

Answer 1

To determine if a matrix is inverrible or not, we need to find the determinant of matrix

If determinant of matrix is zero, then it is invertible, othewrise it non-onvertible

To find the determinant of a matrix, we first note down the values as given and then add the first 2 columns in the next 2 columns

Our given matrix is

0 2 -4

1 0 2

-2 -6 8

The first two columns are

0 2  

1 0

-2 -6  

Now we add the first 2 columns to our matrix, which gives us

0 2 -4 0 2

1 0 2 1 0

-2 -6 8 -2 -6

Now the way to find determinant is

Consider the below one

a1 a2 a3 a1  a2

b1 b2 b3 b1 b2

c1 c2 c3 c1 c2

Now the determinant is given by sbutracting sum of porduct of the diagonals from right to left from sum of porduct of the diagonals from left to right

Here the diagonals from left to right are

(a1 b2 c3) , (a2 b3 c1) , (a3 b1 c2)

Here the diagonals from right to left are

(a3 b2 c1), (a1 b3 c2) , (a2 b1 c3)

So Determinant = (a1*b2 *c3) + (a2*b3*c1) + (a3* b1*c2) - [ (a3*b2*c1) + (a1*b3*c2) - (a2*b1*c3) ]

So determinant for our matrix is

0 2 -4 0 2

1 0 2 1 0

-2 -6 8 -2 -6

Determinant is (0*0*8) + (2*2*-2) + (-4*1*-6) - [ (-4*0*-2) + (0*2*-6) + (2*1*8)]

= 0 + (-8) + 24 - [0 + 0 + 16]

= -8 + 24 -16

= 0

The determinant of the matrix is 0, so it is invertible. So either option A or C is the answer

Now we will convert the matrix to echleon form to know how many pivots are there

0 2 -4  

1 0 2

-2 -6 8

We will replace R3 with R3+2*R2

0 2 -4

1 0 2

0 - 6 12

Now replace R3 with R3 + 3R1

0 2 -4

1 0 2

0 0 0

The first non-zero element in a row is called pivot position

Here 2 is a pivot position in R1 and we don't have any other pivot positions

So the correct Answer is A