Mathematics : Applications of Matrices and Determinants: Summary

**SUMMARY**

(1) Adjoint of a square matrix *A* = Transpose of the
cofactor matrix of *A* .

(2) A(adj A) = (adj A) A = A I_{n}.

(3) A^{-1} = [ 1/|A| ] adj A.

(4) ,

where λ is a non-zero scalar.

(5) (i) (AB)^{-1} = B^{-1} A^{-1} . (ii) ((A)^{-1 })^{-1}=A

(6) If A is a non-singular square matrix of order *n *, then

(8) (i) A matrix A is orthogonal if AA^{T} = A^{T} A = I

(ii) A matrix A is orthogonal if and only if A is non-singular and
A^{−1} = A^{T}

(8) Methods to solve the system of linear equations AX = B

(i) By matrix inversion method X = A^{-1}B, | *A* | ≠ 0

(ii) (ii) By Cramer’s rule

(iii) By Gaussian elimination method

(9) (i) If ρ ( A) = ρ ([ A | B]) = number of unknowns, then the
system has unique solution.

(ii) If ρ ( A) = ρ ([ A | B]) < number of unknowns, then
the system has infinitely many solutions.

(iii) If ρ ( A) ≠ ρ ([ A | B]) then the system is inconsistent and
has no solution.

(10) The homogenous system of linear equations AX = 0

(i) has the trivial solution, if | A | ≠ 0 .

(ii) has a non trivial solution, if | A |= 0 .

Tags : Applications of Matrices and Determinants , 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants : Summary | Applications of Matrices and Determinants

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