badge image

Enroll For Free Now & Improve Your Performance.

×
User Icon
User Icon
User Icon
User Icon
User Icon

Thank you for registering.

One of our academic counsellors will contact you within 1 working day.

Please check your email for login details.
MY CART (5)

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

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping
Menu
Grade: 12th Pass

                        

How to find determinant of a 4th order matrix

8 years ago

Answers : (1)

ruchi yadav
askIITians Faculty
27 Points
							
Any matrixAand its transpose have the same determinant, meaning

\begin{displaymath}\det A = \det A^T.\end{displaymath}

2.
The determinant of a triangular matrix is the product of the entries on the diagonal.
3.
If we interchange two rows, the determinant of the new matrix is the opposite of the old one.
4.
If we multiply one row with a constant, the determinant of the new matrix is the determinant of the old one multiplied by the constant.
5.
If we add one row to another one multiplied by a constant, the determinant of the new matrix is the same as the old one.
6.
We have

\begin{displaymath}\det(AB) = \det(A) \det(B).\end{displaymath}

In particular, ifAis invertible (which happens if and only if$\det(A) \neq 0$), then

\begin{displaymath}\det(A^{-1}) = \frac{1}{\det(A)}.\end{displaymath}

So let us see how this works in case of a matrix of order 4.

Example.Evaluate

\begin{displaymath}\left\vert\begin{array}{cccc}
1&2&3&4\\
5&6&7&8\\
2&6&4&8\\
3&1&1&2\\
\end{array}\right\vert.\end{displaymath}

We have

\begin{displaymath}\left\vert\begin{array}{cccc}
1&2&3&4\\
5&6&7&8\\
2&6&4&8\\...
...3&4\\
5&6&7&8\\
1&3&2&4\\
3&1&1&2\\
\end{array}\right\vert.\end{displaymath}

If we subtract every row multiplied by the appropriate number from the first row, we get

\begin{displaymath}\left\vert\begin{array}{cccc}
1&2&3&4\\
5&6&7&8\\
1&3&2&4\\...
...-4&-8&-12\\
0&1&-1&0\\
0&-5&-8&-10\\
\end{array}\right\vert.\end{displaymath}

We do not touch the first row and work with the other rows. We interchange the second with the third to get

\begin{displaymath}\left\vert\begin{array}{rrrr}
1&2&3&4\\
0&-4&-8&-12\\
0&1&-...
...1&-1&0\\
0&-4&-8&-12\\
0&-5&-8&-10\\
\end{array}\right\vert.\end{displaymath}

If we subtract every row multiplied by the appropriate number from the second row, we get

\begin{displaymath}\left\vert\begin{array}{rrrr}
1&2&3&4\\
0&1&-1&0\\
0&-4&-8&...
...1&-1&0\\
0&0&-12&-12\\
0&0&-13&-10\\
\end{array}\right\vert.\end{displaymath}

Using previous properties, we have

\begin{displaymath}\left\vert\begin{array}{rrrr}
1&2&3&4\\
0&1&-1&0\\
0&0&-12&...
...
0&1&-1&0\\
0&0&1&1\\
0&0&-13&-10\\
\end{array}\right\vert.\end{displaymath}

If we multiply the third row by 13 and add it to the fourth, we get

\begin{displaymath}\left\vert\begin{array}{rrrr}
1&2&3&4\\
0&1&-1&0\\
0&0&1&1\...
...3&4\\
0&1&-1&0\\
0&0&1&1\\
0&0&0&3\\
\end{array}\right\vert\end{displaymath}

which is equal to 3. Putting all the numbers together, we get

\begin{displaymath}\left\vert\begin{array}{cccc}
1&2&3&4\\
5&6&7&8\\
2&6&4&8\\...
...\end{array}\right\vert = 2 \cdot (-1) \cdot (-12) \cdot 3 = 72.\end{displaymath}



Thank You
Ruchi
Askiitians Faculty

6 years ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies


Course Features

  • 731 Video Lectures
  • Revision Notes
  • Previous Year Papers
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Test paper with Video Solution


Course Features

  • 101 Video Lectures
  • Revision Notes
  • Test paper with Video Solution
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Previous Year Exam Questions


Ask Experts

Have any Question? Ask Experts

Post Question

 
 
Answer ‘n’ Earn
Attractive Gift
Vouchers
To Win!!! Click Here for details