
AB=0, if and only if
A . $A\ne 0,B=0$
B . $A=0,B\ne 0$
C . $A=0,B=0$
D . None of these
Answer
162.3k+ views
Hint: To find out which option follows, first we consider the two non- zero matrices A and B.
Matrix are the set of numbers which are arranged in rows and columns to make a rectangular array. In matrix, the numbers are called the entries or the entities of the matrix. Then we multiply A and B matrices and check whether the product gives zero matrix or not. Hence on getting the product of the matrices, we are able to get our correct option.
Complete Step- by- step Solution:
Given $AB=0$
Consider two non- zero matrices A = $\left[\begin{matrix}
0 & 0 \\
0 & 1 \\
\end{matrix} \right]$ and B = $\left[ \begin{matrix}
0 & 1 \\
0 & 0 \\
\end{matrix} \right]$
We see that the both matrices are of $2\times 2$order.
Now we multiply the matrix A with B, we get
AB = $\left[ \begin{matrix}
0 & 0 \\
0 & 1 \\
\end{matrix} \right]$ $\left[ \begin{matrix}
0 & 1 \\
0 & 0 \\
\end{matrix} \right]$
AB = $\left[ \begin{matrix}
0\times 0+0\times 0 & 0\times 1+0\times 0 \\
0\times 0+1\times 0 & 0\times 1+1\times 0 \\
\end{matrix} \right]$
Now on simplifying the above equation, we get
AB = $\left[ \begin{matrix}
0 & 0 \\
0 & 0 \\
\end{matrix} \right]$
Thus, we see that multiplication of two non- zero matrices gives a zero matrix.
Then $A\ne 0,B\ne 0$
Thus, Option ( D) is correct.
Note: In these type of questions, Students take care while multiplying the two matrices. In multiplying the matrices, element of row 1 multiplied with the element of column 1 of another matrix and then we multiply the second element of row 1 with the second element of column 1 of the other matrix. To multiply the two matrices, remember that the number of column of the first matrix match the number of rows of the second matrix.
Matrix are the set of numbers which are arranged in rows and columns to make a rectangular array. In matrix, the numbers are called the entries or the entities of the matrix. Then we multiply A and B matrices and check whether the product gives zero matrix or not. Hence on getting the product of the matrices, we are able to get our correct option.
Complete Step- by- step Solution:
Given $AB=0$
Consider two non- zero matrices A = $\left[\begin{matrix}
0 & 0 \\
0 & 1 \\
\end{matrix} \right]$ and B = $\left[ \begin{matrix}
0 & 1 \\
0 & 0 \\
\end{matrix} \right]$
We see that the both matrices are of $2\times 2$order.
Now we multiply the matrix A with B, we get
AB = $\left[ \begin{matrix}
0 & 0 \\
0 & 1 \\
\end{matrix} \right]$ $\left[ \begin{matrix}
0 & 1 \\
0 & 0 \\
\end{matrix} \right]$
AB = $\left[ \begin{matrix}
0\times 0+0\times 0 & 0\times 1+0\times 0 \\
0\times 0+1\times 0 & 0\times 1+1\times 0 \\
\end{matrix} \right]$
Now on simplifying the above equation, we get
AB = $\left[ \begin{matrix}
0 & 0 \\
0 & 0 \\
\end{matrix} \right]$
Thus, we see that multiplication of two non- zero matrices gives a zero matrix.
Then $A\ne 0,B\ne 0$
Thus, Option ( D) is correct.
Note: In these type of questions, Students take care while multiplying the two matrices. In multiplying the matrices, element of row 1 multiplied with the element of column 1 of another matrix and then we multiply the second element of row 1 with the second element of column 1 of the other matrix. To multiply the two matrices, remember that the number of column of the first matrix match the number of rows of the second matrix.
Recently Updated Pages
If tan 1y tan 1x + tan 1left frac2x1 x2 right where x frac1sqrt 3 Then the value of y is

Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

JEE Energetics Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JoSAA JEE Main & Advanced 2025 Counselling: Registration Dates, Documents, Fees, Seat Allotment & Cut‑offs

NIT Cutoff Percentile for 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

Degree of Dissociation and Its Formula With Solved Example for JEE

Free Radical Substitution Mechanism of Alkanes for JEE Main 2025

JEE Advanced 2025 Notes
