GATE 2026 EE Q.53: solution and answer

GATE 2026 · EEQ.53 · 2 marks · MCQ

A system is represented in state-space form as follows: ( $u$ : input, $x$ : state vector, $y$ : output)

\dot{x} = [1 - 3 20] x + [12] u

y = [12] x

Consider the new state vector $z = [2 - 1 10] x$

What is the state-space representation of the system in terms of the new state vector $z$ ?

(A)
$\dot{z} = [- 1 - 1 4 - 2] z + [4 - 1] u$ , $y = [23] z$
(B)
$\dot{z} = [2033] z + [35] u$ , $y = [23] z$
(C)
$\dot{z} = [4 - 2 9 - 3] z + [4 - 1] u$ , $y = [23] z$
(D)
$\dot{z} = [2 - 4 11] z + [4 - 1] u$ , $y = [4 - 1] z$

See the question in the paper

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Answer and solution

ANSWER

(C): $\dot{z} = [4 - 2 9 - 3] z + [4 - 1] u$ , $y = [23] z$

Official answer: CSource: Official answer key, GATE 2026 EE (IIT Guwahati)

Similarity transformation: with $z = T x$ , $T = [2 - 1 10]$ , we have $x = T^{- 1} z$ . Substituting into $\dot{x} = A x + B u$ and pre-multiplying by $T$ gives $\dot{z} = T A T^{- 1} z + T B u$ , and $y = C x = C T^{- 1} z$ .
Inverse of $T$ : $det T = (2) (0) - (1) (- 1) = 1$ , so $T^{- 1} = [01 - 1 2]$ . Check: $T T^{- 1} = [1001]$ .
Input matrix: $T B = [2 - 1 10] [12] = [4 - 1]$ . This rules out (B).
Output matrix: $C T^{- 1} = [12] [01 - 1 2] = [23]$ . This rules out (D).
System matrix: first $A T^{- 1} = [1 - 3 20] [01 - 1 2] = [2033]$ , then $T A T^{- 1} = [2 - 1 10] [2033] = [4 - 2 9 - 3]$
Invariance check: a similarity transform keeps the trace and the determinant. $tr A = 1 + 0 = 1$ and $det A = (1) (0) - (2) (- 3) = 6$ . For the new matrix, $4 + (- 3) = 1$ and $(4) (- 3) - (9) (- 2) = 6$ . Both match. The matrix in (A) has trace $- 3$ , (B) has trace $5$ and (D) has trace $3$ , so none of them can be similar to $A$ .
So $\dot{z} = [4 - 2 9 - 3] z + [4 - 1] u$ , $y = [23] z$ , which is option (C).

KEY IDEA

For a change of state $z = T x$ , the new model is $(T A T^{- 1}, T B, C T^{- 1}, D)$ . The eigenvalues, and so the trace and determinant, of the system matrix do not change. That gives a quick check that removes wrong options without doing all the multiplication.

WATCH OUT FOR

Forgetting to pre-multiply by $T$ and stopping at $A T^{- 1} = [2033]$ gives the system matrix printed in (B). Mixing up the direction of the transform is the other trap. Because the question defines $z = T x$ , the new matrix is $T A T^{- 1}$ , not $T^{- 1} A T$ . The two have the same trace and determinant, so the invariance check cannot catch this mistake. Only reading the definition carefully does. The same applies to the output matrix: it is $C T^{- 1}$ , not $C T$ .

From the GATE 2026 EE paper. Every question of this paper · Download the full paper · All 16 State space questions, with notes

GATE 2026 EE Q.53: solution and answer

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