The inverse of the matrix
(
1

0

1

4

2

0

𝑎

0

𝑎
2
)
is
(
−
1

0

0.5

2

0.5

−
1

𝑎
2

0

−
0.5
)
.

What is
𝑎
?

Question

The inverse of the matrix 
(
1
	
0
	
1

4
	
2
	
0

𝑎
	
0
	
𝑎
2
)
is 
(
−
1
	
0
	
0.5

2
	
0.5
	
−
1

𝑎
2
	
0
	
−
0.5
)
.

What is 
𝑎
?

BlackTom AI · Accepted Answer

We need to determine the value(s) of a for which the given matrix A and its claimed inverse B satisfy A * B = I.

First, restate the data clearly:
- A = [ [1, 0, 1], [4, 2, 0], [a, 0, a^2] ]
- B = [ [-1, 0, 0.5], [2, 0.5, -1], [a^2, 0, -0.5] ]
- The options for a are presented as:
  • "1 4" (two possibilities, 1 or 4)
  • "None of the other options"
  • "2"
  • "0"
  • "1 2" (two possibilities, 1 or 2)
  • "1"

We verify the (1,1) entry of the product A * B, since that yields a direct equation for a:
- (A * B)_{11} = row1 of A dotted with column1 of B = [1, 0, 1] · [-1, 2, a^2] = -1 + a^2.
- For A * B to be the identity, (A * B)_{11} must equal 1. Thus we require -1 + a^2 = 1, which gives a^2 = 2.
- This would imply a = sqrt(2) or a = -sqrt(2).

Now test these against another independent entry to see consistency. Take (A * B)_{33}:
- (A * B)_{33} = row3 of A dotted with column3 of B = [a, 0, a^2] · [0.5, -1, -0.5] = 0.5a - 0.5a^2.
- With a^2 = 2 from above, this becomes 0.5a - 0.5*2 = 0.5a - 1.
- For the identity matrix, (A * B)_{33} must be 1, so we require 0.5a - 1 = 1, i.e., 0.5a = 2, giving a = 4.

We now have a contradiction: from the (1,1) entry we must have a^2 = 2 (a = ±√2), but from the (3,3) entry we must have a = 4. There is no value of a that simultaneously satisfies both conditions. This means no value of a makes A * B equal to the identity matrix given by the provided B.

To illustrate this with the provided options:
- If a = 1 (the two-number option "1 4" includes 1), then (A * B)_{11} = -1 + 1 = 0, not 1, so it fails.
- If a = 4 (the same option), then (A * B)_{11} = -1 + 16 = 15, not 1, so it fails.
- If a = 2 (option "2" or the "1 2" group), then (A * B)_{11} = -1 + 4 = 3, not 1, so it fails.
- If a = 0 (option "0"), then (A * B)_{11} = -1 + 0 = -1, not 1, so it fails.
- If a = 1 (option "1"), same as the first case, it fails.
- The only remaining option explicitly asserting a nonexistence, "None of the other options", is consistent with there being no a that satisfies all entries of A * B = I.

In summary, none of the listed a values satisfy the necessary condition from the (1,1) entry, and the implied additional condition from the (3,3) entry leads to a further contradiction. Consequently, there is no value of a among the given options that yields an exact inverse as specified.

The inverse of the matrix ( 1 0 1 4 2 0 𝑎 0 𝑎 2 ) is ( − 1 0 0.5 2 0.5 − 1 𝑎 2 0 − 0.5 ) . What is 𝑎 ?

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