Click to Chat

0120-4616500

CART 0

• 0

MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: R

There are no items in this cart.
Continue Shopping
Get instant 20% OFF on Online Material.
coupon code: MOB20 | View Course list

Get extra R 550 off
USE CODE: SSPD25

				   what is proof of
where A is a square matrix of order n and I is identity matrix of order n. plz reply with proper derivation.



6 years ago

Share

### Answers : (1)

										Dear vikash,
As a consequence of Laplace formula for the determinant of an n×n matrix A, we have
$\mathbf{A}\, \mathrm{adj}(\mathbf{A}) = \mathrm{adj}(\mathbf{A})\, \mathbf{A} = \det(\mathbf{A})\, \mathbf{I}\qquad (*)$
where I is the n×n identity matrix Indeed, the (i,i) entry of the product A adj(A) is the scalar product of row i of A with row i of the cofactor matrix C, which is simply the Laplace formula for det(A) expanded by row i. Moreover, for i ≠ j the (i,j) entry of the product is the scalar product of row i of A with row j of C, which is the Laplace formula for the determinant of a matrix whose i and j rows are equal and is therefore zero.
From this formula follows one of the most important results in matrix algebra: A matrix A over a commutative ring R is invertible if and only if det(A) is invertible in R.
For if A is an invertible matrix then
$1 = \det(\mathbf I) = \det(\mathbf{A} \mathbf{A}^{-1}) = \det(\mathbf{A}) \det(\mathbf{A}^{-1}),$
and if det(A) is a unit then (*) above shows that
$\mathbf{A}^{-1} = \det(\mathbf{A})^{-1}\, \mathrm{adj}(\mathbf{A}).$
We are all  IITians and here to help you in your IIT  JEE preparation.  All the best.
If you like this answer please approve it....
win exciting gifts by answering   the questions on Discussion Forum

Sagar Singh
B.Tech IIT Delhi


6 years ago

# Other Related Questions on Algebra

what are the solutions of following equation? Ix 3 -6x 2 +11x-6I

Through trial an error method we can see that (x-1) is a factor. Then by vanishing method the polynomial can be solved as :f(x) = x^3-6x^2+11x-6= x^3-x^2-5x^2+5x+6x-6= x^2(x-1) -5x(x-1)...

 Shaswata Biswas one month ago
no of sol. of Ix+2I+IxI+Ix-2I=p ,p belongs to Real no. are?

 jagdish singh singh one month ago

draw the graph of given eq. hence getting if p=4 we getone sol.i.e x=0 while for all p>4 two solutions.

 Pranav Dabhade one month ago
modulus operator is a bit tough for me and i cant understand a thing in some questions for eg lim x->0- mod(x)/x=-1; how does this work mod must always be positive right?

Posting again as the there is bug in website which does not properly dispay inequalities. Its not difficult if you visualise it graphically. Search google for desmos graphing calculator and ...

 Ajay 2 months ago

Its not difficult if you visualise it graphically. Search google for desmos graphing calculator and plot the graph for this function y= mod(x)/x. You will see why left hand limit for thins...

 Ajay 2 months ago
Do vectors satisfy associative law over addition????????

yes the vectors satisfies associative law over addition................................................................

 sai hemanth kumar 4 months ago

yes the vectors satisfy the associative law over addition because (a+b)+c=a+(b+c) and vectors will not satify the associative law over multiplication

 SREEKANTH 4 months ago

yes the vectors satisfies associative law over addition................................................................

 Poosala Vedaprakash 4 months ago
It shows that it is in upsc 2013 if any were there of that batch pls verify that was this question there r not and plz solve it.

HEY DEAR There is no solution to this question using normal methods, as three odd numbers added together will always produce an odd number.Since the answer is even, and an odd number can...

 SAHIL one month ago

WE PROVIDE 24 X 7 SUPPORT TO EVERYONE. WHENEVER A DOUBT WHETHER IS IS RELATED WITH HOW TO PREPARE, WHETHER IT IS HOW TO START, WHICH BOOKS TO BUY ANYTHING YOU ARE FREE TO ASK. IF YOU WANT...

 SAHIL one month ago

SORRY DEAR, BUT I DONOT USE WHATSAPP AS IT IS COMPLETE WASTAGE OF YOUR TIME.I THINK IF YOU ARE A UPSC ASPIRANT THEN YOU SHOULD ALSO LEAVE THESE SOCIAL SITES REMEMBER.” GREAT MINDS DISCUSS...

 SAHIL one month ago
View all Questions »

• Complete JEE Main/Advanced Course and Test Series
• OFFERED PRICE: R 15,000
• View Details
Get extra R 3,750 off
USE CODE: SSPD25

Get extra R 550 off
USE CODE: SSPD25

More Questions On Algebra