(B) Statement $(i)$: $AB=0$ does not necessarily imply $A=0$ or $B=0$. For example,consider two non-zero matrices $A = \begin{bmatrix} 0 & 1 \\ 0 & 0

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(ii) is true and $(i)$,(iii) are false

(B) Statement $(i)$: $AB=0$ does not necessarily imply $A=0$ or $B=0$. For example,consider two non-zero matrices $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$. Their product is $AB = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = 0$. Thus,$(i)$ is false.Statement (ii): If $AB=I_3$,then by the definition of an inverse matrix,$B$ is the inverse of $A$ (i.e.,$A^{-1}=B$). Thus,(ii) is true.Statement (iii): The expansion $(A-B)^2 = (A-B)(A-B) = A^2 - AB - BA + B^2$. Since matrix multiplication is not commutative in general $(AB \neq BA)$,$A^2 - AB - BA + B^2$ is not equal to $A^2 - 2AB + B^2$ unless $AB=BA$. Thus,(iii) is false.Therefore,only (ii) is true.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Mix Examples-Determinants and Matrices

Medium

If $A$ and $B$ are both $3 \times 3$ matrices,then which of the following statements are true?
$(i)$ $AB=0 \Rightarrow A=0$ or $B=0$
(ii) $AB=I_3 \Rightarrow A^{-1}=B$
(iii) $(A-B)^2=A^2-2AB+B^2$

AP EAMCET 2025 All AP EAMCET

A
$(i)$ is false and (ii),(iii) are true
B
(ii) is true and $(i)$,(iii) are false
C
$(i)$ and (ii) are true,(iii) is false
D
All are true

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3 and 4 .Determinants and Matrices

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Mix Examples-Determinants and Matrices

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Let $S = \{\sqrt{n} : 1 \leq n \leq 50, n \text{ is odd}\}$. Let $a \in S$ and $A = \begin{bmatrix} 1 & 0 & a \\ -1 & 1 & 0 \\ -a & 0 & 1 \end{bmatrix}$. If $\sum_{a \in S} \operatorname{det}(\operatorname{adj} A) = 100 \lambda$,then $\lambda$ is equal to:

MediumJEE MAIN 2022

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Consider the following linear equations:
$ax+by+cz=0$,$bx+cy+az=0$,$cx+ay+bz=0$
Match the conditions/expressions in Column $I$ with statements in Column $II$:
Column $I$ Column $II$
$(A)$ $a+b+c \neq 0$ and $a^2+b^2+c^2=ab+bc+ca$ $(p)$ The equations represent planes meeting only at a single point.
$(B)$ $a+b+c=0$ and $a^2+b^2+c^2 \neq ab+bc+ca$ $(q)$ The equations represent the line $x=y=z$.
$(C)$ $a+b+c \neq 0$ and $a^2+b^2+c^2 \neq ab+bc+ca$ $(r)$ The equations represent identical planes.
$(D)$ $a+b+c=0$ and $a^2+b^2+c^2=ab+bc+ca$ $(s)$ The equations represent the whole of the three-dimensional space.

IIT 2007

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The number of $3 \times 2$ matrices $A$,which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^{T}A$ is $5$,is . . . . . . .

DifficultJEE MAIN 2026

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If $\omega$ is a root of the equation $x+\frac{1}{x}+1=0$,then the value of the determinant $\left|\begin{array}{ccc}1 & 1+\omega & 1+\omega+\omega^2 \\ 3 & 4+3 \omega & 5+4 \omega+3 \omega^2 \\ 6 & 9+6 \omega & 11+9 \omega+6 \omega^2\end{array}\right|$ is equal to

DifficultAP EAMCET 2021

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Consider the following relation $R$ on the set of real square matrices of order $3$. $R = \{(A,B) | A = P^{-1}BP \text{ for some invertible matrix } P\}$.
\textbf{Statement-$1$:} $R$ is an equivalence relation.
\textbf{Statement-$2$:} For any two invertible $3 \times 3$ matrices $M$ and $N$,$(MN)^{-1} = N^{-1}M^{-1}$.

MediumAIEEE 2011

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Column $I$	Column $II$
$(A)$ $a+b+c \neq 0$ and $a^2+b^2+c^2=ab+bc+ca$	$(p)$ The equations represent planes meeting only at a single point.
$(B)$ $a+b+c=0$ and $a^2+b^2+c^2 \neq ab+bc+ca$	$(q)$ The equations represent the line $x=y=z$.
$(C)$ $a+b+c \neq 0$ and $a^2+b^2+c^2 \neq ab+bc+ca$	$(r)$ The equations represent identical planes.
$(D)$ $a+b+c=0$ and $a^2+b^2+c^2=ab+bc+ca$	$(s)$ The equations represent the whole of the three-dimensional space.

If $A$ and $B$ are both $3 \times 3$ matrices,then which of the following statements are true?$(i)$ $AB=0 \Rightarrow A=0$ or $B=0$(ii) $AB=I_3 \Rightarrow A^{-1}=B$(iii) $(A-B)^2=A^2-2AB+B^2$

Similar Questions

Let $S = \{\sqrt{n} : 1 \leq n \leq 50, n \text{ is odd}\}$. Let $a \in S$ and $A = \begin{bmatrix} 1 & 0 & a \\ -1 & 1 & 0 \\ -a & 0 & 1 \end{bmatrix}$. If $\sum_{a \in S} \operatorname{det}(\operatorname{adj} A) = 100 \lambda$,then $\lambda$ is equal to:

The number of $3 \times 2$ matrices $A$,which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^{T}A$ is $5$,is . . . . . . .

If $\omega$ is a root of the equation $x+\frac{1}{x}+1=0$,then the value of the determinant $\left|\begin{array}{ccc}1 & 1+\omega & 1+\omega+\omega^2 \\ 3 & 4+3 \omega & 5+4 \omega+3 \omega^2 \\ 6 & 9+6 \omega & 11+9 \omega+6 \omega^2\end{array}\right|$ is equal to

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If $A$ and $B$ are both $3 \times 3$ matrices,then which of the following statements are true?
$(i)$ $AB=0 \Rightarrow A=0$ or $B=0$
(ii) $AB=I_3 \Rightarrow A^{-1}=B$
(iii) $(A-B)^2=A^2-2AB+B^2$