Properties Matrix

If A, B, C are m × n matrix and c and d are scalars, the following properties are true.

(1) commutative property with respect to addition

A+B= B+A

(2) associative property with respect to addition

(A+B)+C= A+(B+C)

(3) associative property of scalar addition

(c d) A=c (d A) where c and d are scalar.

(4) Existence of additive identity OR null matrix

A+0= 0+A =A (where 0 is null matrix)

(5) Existence of multiplicative identity(unit/identity or imaginary matrix)

1A= A1= A

(6) Distributive property w.r.t scalar multiplication.

(a) c (A+B) = c A + c. B

(b) (c+d) A= c A + d A

(7) associative property with respect to multiplication.

A(BC) = (AB)C

(8) Left distribution Law (property)

A(B+C)= AB+AC

(9) Right distribution property

(A+B)C = AC+BC

(10) Right scalar distribution property.

c (AB) = (c A)B= A(c B)

(11) diagonal matrix properties

A diagonal matrix is called an identity matrix. If all diagonal entries are 1

EXAMPLE

EQUAL MATRIX:

any two matrices A and B are called equal matrix, if

(i) order of A = order of B

(ii) when corresponding entries are same

ADDITIVE INVERSE MATRIX:

Let A be a matrix. A matrix B is defined as an additive inverse of A. If

B+A=0= A+B

DETERMINANT OF M:

Let $property matrix in determinant 2by 2$

A real number (lambda) is called determinant of M, denoted by M

ADJOINT OF M MATRIX:

for a matrix M= $Adjoint of matrix 2 by 2$

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