JEE Main & Advanced Mathematics Determinants & Matrices Scalar Multiplication of Matrices

 Let \[A={{[{{a}_{ij}}]}_{m\times n}}\]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 \[A={{[{{a}_{ij}}]}_{m\times n}}\], then \[kA=Ak={{[k{{a}_{ij}}]}_{m\times n}}\].

Properties of scalar multiplication

If A, B are matrices of the same order and \[\lambda ,\,\mu \] are any two scalars then

(i) \[\lambda (A+B)=\lambda A+\lambda B\]                     

(ii) \[(\lambda +\mu )A=\lambda A+\mu A\]

(iii) \[\lambda (\mu A)=(\lambda \mu A)=\mu (\lambda A)\]

(iv) \[(-\lambda A)=-(\lambda A)=\lambda \,(-A)\]

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