Matrix Multiplication Compatibility: The Inner-Dimensions Rule
You cannot multiply just any two matrices. Multiplication is defined only when their sizes line up in a specific way, and this compatibility check is always the very first step of any product.
The Idea: A Row Meets a Column
When you multiply A by B, you take each row of A and pair it against each column of B, multiply corresponding entries, and add. For that pairing to work, a row of A must have exactly as many entries as a column of B.
Note
Think of a handshake: each finger of one hand must meet a finger of the other. If one hand has 4 fingers and the other has 3, the handshake fails.
The Precise Statement
Let A be m×n and B be p×q.
A×B is defined if and only if n=p — the number of columns of A equals the number of rows of B. The product C=AB then has order m×q.
Writing the sizes side by side, (m×n)(p×q), the inner numbers (n,p) must match; the outer numbers (m,q) give the result's shape.
Why the Rule Exists
Each entry of the product is
cij=∑k=1naikbkj.
Here k runs over the columns of A (up to n) and the rows of B (up to p). If n=p, the sum runs out of matching terms and is meaningless — that is exactly why compatibility demands n=p.
Watch out
Even when both AB and BA are defined, they usually differ. For A of order 2×3 and B of order 3×2, AB is 2×2 but BA is 3×3 — different sizes entirely. Matrix multiplication is not commutative.
Quick Check
A
B
Defined?
Result
2×3
3×4
Yes
2×4
2×3
2×4
No
—
1×4
4×1
Yes
1×1
Tip
Before multiplying, write both orders side by side and circle the inner numbers. If they are equal, multiply; if not, stop — the product does not exist.
Matrix Multiplication Compatibility — the rule that the number of columns of the first matrix must equal the number of rows of the second — is one of the first checks taught in the CBSE Class 12 Matrices chapter, and "matrix multiplication rules class 12" is a common search among students preparing for board exams and JEE Main. NCERT's own solved examples emphasize checking this condition before attempting any product.
Concept: Distributive property of matrix multiplication over addition.
AC+BC=91230+18−2=102028, which matches (A+B)C.
✓Final answer
The distributive property is verified: (A+B)C=AC+BC=102028.
The problem tests the distributive property of matrix multiplication over addition. We compute AC, BC, and (A+B)C directly, and verify that (A+B)C=AC+BC holds exactly.
The core idea here is the distributive property of matrix multiplication: for matrices of compatible sizes,
(A+B)C=AC+BC. This is not just a rule to memorise — it follows from the fact that matrix multiplication is defined entry‑wise as a sum of products, and addition of matrices is entry‑wise. So when you multiply a sum of matrices by a column vector, each entry in the result is a sum of two separate dot products, which can be rearranged.
We are given three matrices: A (a 3×3 skew‑symmetric matrix), B (another 3×3 matrix), and C (a 3×1 column vector). All multiplications are defined because the number of columns in A and B (3) matches the number of rows in C (3).
A common mistake is to forget that matrix addition must be done before multiplication when computing (A+B)C — you cannot multiply A and B separately by C and then add the matrices unless you are using the distributive property, which we are verifying here. The order matters: (A+B)C means add first, then multiply.
Tip
The distributive property works because matrix multiplication is linear in the left factor. This is the same reason that (A+B)C=AC+BC always holds when the dimensions are compatible — it’s not a coincidence, it’s built into the definition.
✓Final answer
The computed values are AC=91230, BC=18−2, (A+B)C=102028, and we have verified that (A+B)C=AC+BC.
Method: Verifying the distributive law (A+B)C=AC+BC
Use this when asked to confirm distribution of matrix multiplication over addition.
Steps
Step 1: Compute the left side by adding first, then multiplying.
Form A+B entry-wise, then multiply by C (row-by-column).
Step 2: Compute the right side by multiplying first, then adding.
Find AC and BC separately, then add them entry-wise.
Step 3: Compare the two results.
They must be identical, confirming distributivity.
Common Mistakes
Mistake 1: Computing (A+B)C as AC⋅BC or some product.
Why it's wrong: distribution gives a sumAC+BC, not a product of the two. Correct approach: add AC and BC.
Mistake 2: Adding A and B after multiplying by C on the left side incorrectly.
Why it's wrong: (A+B)C means add A,B first, then multiply once by C. Correct approach: respect the grouping — one multiplication on the left, two-then-add on the right.
Mistake 3: Sign errors in the column-vector dot products.
Why it's wrong: entries like (6)(−2)+(7)(3) need careful signs. Correct approach: track the sign of every term.
The expression A2+B(A+B) simplifies to (A+B)2−AB by using the distributive property of matrix multiplication. We calculate (A+B)2 and then subtract AB to find the result 431646623.
The core idea here is to simplify the given expression A2+B(A+B) using the properties of matrix algebra, specifically the distributive property of matrix multiplication over addition. We are given A+B and AB, so we should try to express the target expression in terms of these known quantities.
First, let's expand B(A+B):
B(A+B)=BA+B2
So, the expression we need to evaluate becomes:
A2+B(A+B)=A2+BA+B2
Now, let's recall the expansion of (A+B)2 for matrices:
For matrices A and B, (A+B)2=(A+B)(A+B)=A(A+B)+B(A+B)=A2+AB+BA+B2.
Watch out
It is crucial to remember that matrix multiplication is generally not commutative, meaning AB=BA. Therefore, (A+B)2 is not equal to A2+2AB+B2 unless AB=BA.
Comparing the expression we need, A2+BA+B2, with the expansion of (A+B)2, which is A2+AB+BA+B2, we can see a direct relationship.
If we subtract AB from (A+B)2, we get:
(A+B)2−AB=(A2+AB+BA+B2)−AB
=A2+(AB−AB)+BA+B2
=A2+0+BA+B2
=A2+BA+B2
This shows that A2+B(A+B)=(A+B)2−AB. This simplification is key because we are given A+B and AB directly.
Now, we can proceed with the calculations:
Calculate (A+B)2:
We are given A+B=210122202.
(A+B)2=(A+B)(A+B)=210122202210122202
Let's perform the matrix multiplication:
First row:
(2)(2)+(1)(1)+(2)(0)=4+1+0=5
(2)(1)+(1)(2)+(2)(2)=2+2+4=8
(2)(2)+(1)(0)+(2)(2)=4+0+4=8
Second row:
(1)(2)+(2)(1)+(0)(0)=2+2+0=4
(1)(1)+(2)(2)+(0)(2)=1+4+0=5
(1)(2)+(2)(0)+(0)(2)=2+0+0=2
Third row:
(0)(2)+(2)(1)+(2)(0)=0+2+0=2
(0)(1)+(2)(2)+(2)(2)=0+4+4=8
(0)(2)+(2)(0)+(2)(2)=0+0+4=4
So, (A+B)2=542858824.
Subtract AB from (A+B)2:
We are given AB=111212201.
Now, we compute (A+B)2−AB:
A2+B(A+B)=542858824−111212201
=5−14−12−18−25−18−28−22−04−1
=431646623
Comparing this result with the given options, it matches option (A).