## How to Multiply Matrices

### Multiplication of matrices

Two matrices A and B are conformable for the product AB if the number of columns in A (pre-multiplier) is same as the number of rows in B (post multiplier). Thus, if A = [a_{ij}]_{m×n }and B = [b_{ij}]_{n×p } are two matrices of order m × n and n × p respectively, then their product AB is of order and is defined as

Thus, from (i), (AB)_{ij} = Sum of the product of elements of i^{th} row of A with the corresponding elements of j^{th} column of B.

### Properties of matrix multiplication

If A,B and C are three matrices such that their product is defined, then

- AB ≠ BA, (Generally not commutative)
- (AB)C = A(BC), (Associative Law)
- IA = A = AI, where I is identity matrix for matrix multiplication.
- A(B + C) = AB + AC, (Distributive law)
- If AB = AC ⇏ B = C, (Cancellation law is not applicable)
- If AB = 0, it does not mean that A = 0 or B = 0, again product of two non zero matrix may be a zero matrix.

### Scalar multiplication of matrices

Let A = [a_{ij}]_{m×n} be a matrix and k be a number, then the matrix which is obtained by multiplying every element of A by k is called scalar multiplication of A by k and it is denoted by kA.

Thus, if A = [a_{ij}]_{m×n}, then kA = Ak = [ka_{ij}]_{m×n}.

**Properties of scalar multiplication:**

If A, B are matrices of the same order and λ, μ are any two scalars then

- λ (A + B) = λA + λB
- (λ + μ)A = λ + μA
- λ(μA) = (λμA) = μ(λA)
- (-λA) = -(λA) = λ(-A)

All the laws of ordinary algebra hold for the addition or subtraction of matrices and their multiplication by scalars.

### Multiplication of Matrices Problems with Solutions

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