
What are the properties of matrix multiplication?
Answer
162k+ views
Hints Recall the formation of a matrix that is how a matrix looks like, then write the properties of matrix multiplication.
Complete step by step solution
Example of a matrix is \[\left[ {\begin{array}{*{20}{c}}a&b&c\\d&e&f\\g&h&i\end{array}} \right]\] .
Now, suppose two matrices are \[\left[ {\begin{array}{*{20}{c}}a&b&c\\d&e&f\\g&h&i\end{array}} \right]\]and \[\left[ {\begin{array}{*{20}{c}}p&q&r\\s&t&u\\v&w&x\end{array}} \right]\].
Hence, the multiplication of two matrices is,
\[\left[ {\begin{array}{*{20}{c}}a&b&c\\d&e&f\\g&h&i\end{array}} \right].\left[ {\begin{array}{*{20}{c}}p&q&r\\s&t&u\\v&w&x\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ap + bs + cv}&{aq + bt + cw}&{ar + bu + cx}\\{dp + es + fv}&{dq + et + fw}&{dr + eu + fx}\\{gp + hs + iv}&{gq + ht + iw}&{gr + hu + ix}\end{array}} \right]\]
The properties of a matrix multiplication are,
(1) Associative law: If A, B and C are non-zero matrices then \[(AB)C = A(BC)\] .
(2) Distributive law: If A, B and C are non-zero matrices, then \[A.(B + C) = A.B + A.C\]
Or, \[(B + C).A = B.A + C.A\].
(3) Multiplicative identity property: If I be the identity matrix then \[A.I = I.A = A\] .
The properties of a matrix multiplication are associative law, distributive law and the identity property.
Additional information:
The matrix multiplication does not follow commutative property. Suppose the order of A is \[m\times n\] and the order of B is \[i \times j\]. Then matrix multiplication of AB exists if i = n.
Note Sometime students give the proofs with the properties, but that is not needed as the question is asking only to mention the properties of matrix multiplication. Only you can do that show the matrix multiplication but not elaborately just state the multiplication and then go for the properties of matrix multiplication as per the requirement of the question.
Complete step by step solution
Example of a matrix is \[\left[ {\begin{array}{*{20}{c}}a&b&c\\d&e&f\\g&h&i\end{array}} \right]\] .
Now, suppose two matrices are \[\left[ {\begin{array}{*{20}{c}}a&b&c\\d&e&f\\g&h&i\end{array}} \right]\]and \[\left[ {\begin{array}{*{20}{c}}p&q&r\\s&t&u\\v&w&x\end{array}} \right]\].
Hence, the multiplication of two matrices is,
\[\left[ {\begin{array}{*{20}{c}}a&b&c\\d&e&f\\g&h&i\end{array}} \right].\left[ {\begin{array}{*{20}{c}}p&q&r\\s&t&u\\v&w&x\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ap + bs + cv}&{aq + bt + cw}&{ar + bu + cx}\\{dp + es + fv}&{dq + et + fw}&{dr + eu + fx}\\{gp + hs + iv}&{gq + ht + iw}&{gr + hu + ix}\end{array}} \right]\]
The properties of a matrix multiplication are,
(1) Associative law: If A, B and C are non-zero matrices then \[(AB)C = A(BC)\] .
(2) Distributive law: If A, B and C are non-zero matrices, then \[A.(B + C) = A.B + A.C\]
Or, \[(B + C).A = B.A + C.A\].
(3) Multiplicative identity property: If I be the identity matrix then \[A.I = I.A = A\] .
The properties of a matrix multiplication are associative law, distributive law and the identity property.
Additional information:
The matrix multiplication does not follow commutative property. Suppose the order of A is \[m\times n\] and the order of B is \[i \times j\]. Then matrix multiplication of AB exists if i = n.
Note Sometime students give the proofs with the properties, but that is not needed as the question is asking only to mention the properties of matrix multiplication. Only you can do that show the matrix multiplication but not elaborately just state the multiplication and then go for the properties of matrix multiplication as per the requirement of the question.
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
