Algebra

Matrices & Determinants

Medium

applied

For the system of equations AX = B, which of the following is correct?

If A is a singular matrix, then AX = B has a unique solution.

If A is a non-singular matrix, then AX = B has infinitely many solutions.

If A is a non-singular matrix, then AX = B has a unique solution.

If A is a singular matrix, then AX = B is always consistent.

✅ Correct Option: 3

For the system of equations $AX = B$ :

A non-singular matrix has determinant $\neq 0$ and an inverse $A^{-1}$ exists.

A singular matrix has determinant $= 0$ and no inverse exists.

When $A$ is non-singular, the system has a unique solution.

Since $A^{-1}$ exists, multiply both sides by $A^{-1}$ :

$A^{-1}(AX) = A^{-1}B$

$(A^{-1}A)X = A^{-1}B$

$X = A^{-1}B$

This gives exactly one solution.

When $A$ is singular, the system cannot have a unique solution.

Since $A^{-1}$ does not exist, the system may have either infinitely many solutions or no solution at all, depending on $B$ .

Checking Statement 1: "If $A$ is singular, then $AX = B$ has a unique solution."

This is incorrect. A singular matrix cannot produce a unique solution.

Checking Statement 2: "If $A$ is non-singular, then $AX = B$ has infinitely many solutions."

This is incorrect. A non-singular matrix produces exactly one solution $X = A^{-1}B$ , not infinitely many.

Checking Statement 3: "If $A$ is non-singular, then $AX = B$ has a unique solution."

This is correct. When $A$ is non-singular, $X = A^{-1}B$ gives a unique solution.

Checking Statement 4: "If $A$ is singular, then $AX = B$ is always consistent."

This is incorrect. A singular matrix does not guarantee consistency; the system may have no solution depending on $B$ .

Therefore, the correct answer is Option 3.

2026: 30 May Shift 2

If A is square matrix of order 3 x 3 such that $a_{ij} = j - i$ , then which of the following statements are correct?

(A) A is a symmetric matrix.

(B) A is a skew symmetric matrix.

(D) A is a diagonal matrix.

Choose the correct answer from the options given below:

2026: 12 May Shift 2

If $\begin{vmatrix}4x & 9\\ -4 & -5\end{vmatrix}=\begin{vmatrix}4 & 8\\ 2 & 3\end{vmatrix}$ , then the value of $x$ is :

2026: 25 May Shift 1

If $A = \begin{bmatrix} 1 & a \\ b & 2 \end{bmatrix}$ and $B = \begin{bmatrix} a & 1 \\ 2 & -3b \end{bmatrix}$ such that $\lvert AB\rvert = \lvert 2B\rvert$ and $ab \neq -\dfrac{2}{3}$ then:

A. $\lvert A\rvert = 2$

B. $\lvert A\rvert = 4$

C. $\lvert B\rvert = 2$

D. $\lvert B\rvert = 4$

Choose the correct answer from the options given below:

2022: 10 Aug Shift 1

The length ( $x$ ) and breadth ( $y$ ) of plot satisfy equations:

2022: 17 Aug Shift 2

Identify the correct statements.

(A) If $A$ is a non-singular matrix, then $A^{-1} = \frac{|A|}{(\text{adj } A)}$

(B) If $A$ is an invertible matrix then $\frac{1}{|A^{-1}|} = |A|$

(C) If $A$ and $B$ are two invertible matrices of the same order then $AB$ is also invertible matrix and $(BA)^{-1} = A^{-1}B^{-1}$

(D) If $A$ is an invertible matrix, then $A^T$ is also invertible and $(A^T)^{-1} = \frac{1}{(A^{-1})^T}$

Choose the correct answer from the options given below :

2026: 19 May Shift 2

The value of $\begin{vmatrix} \log_3 512 & \log_4 3 \\ \log_3 8 & \log_4 9 \end{vmatrix}$ is:

For the system of equations AX = B, which of the following is correct?

Solution

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