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Calculation of Values of Determinants upto Third Order - Business Mathematics & Statistics Video Lecture | Business Mathematics and Statistics - B Com
FAQs on Calculation of Values of Determinants upto Third Order - Business Mathematics & Statistics Video Lecture - Business Mathematics and Statistics - B Com
| 1. How do you calculate the value of a determinant? |  |
Ans. To calculate the value of a determinant, you need to follow these steps:
1. Determine the order of the determinant (e.g., second order or third order).
2. Write down the elements of the determinant in a square matrix format.
3. Multiply the elements of the main diagonal from top left to bottom right.
4. Multiply the elements of the secondary diagonal from top right to bottom left.
5. Subtract the product of the secondary diagonal from the product of the main diagonal.
6. The result obtained is the value of the determinant.
| 2. What is the significance of the determinant in business mathematics and statistics? | |
Ans. Determinants have various applications in business mathematics and statistics. They are used to solve systems of linear equations, find the inverse of a matrix, calculate areas and volumes, and determine the consistency of a system of linear equations. In business, determinants help in decision-making processes, risk analysis, and optimization problems.
| 3. Can you give an example of calculating a determinant of third order? | |
Ans. Sure! Let's take the following matrix as an example:
| 2 4 6 |
| 1 3 5 |
| 7 9 8 |
To calculate the determinant, we follow these steps:
1. Multiply the elements of the main diagonal: 2 * 3 * 8 = 48
2. Multiply the elements of the secondary diagonal: 6 * 3 * 7 = 126
3. Subtract the product of the secondary diagonal from the product of the main diagonal: 48 - 126 = -78
So, the value of the determinant for this matrix is -78.
| 4. How are determinants useful in solving systems of linear equations? | |
Ans. Determinants play a crucial role in solving systems of linear equations. By representing the coefficients of the variables in a matrix, we can calculate the determinant of the matrix. If the determinant is non-zero, it indicates that the system of equations has a unique solution. If the determinant is zero, it means the system has either no solution or infinite solutions. Determinants help us determine the consistency and compatibility of a system of linear equations.
| 5. Is there a shortcut method to calculate determinants of higher order matrices? | |
Ans. Yes, there is a shortcut method called the Laplace expansion or cofactor expansion. In this method, you choose any row or column of the matrix and multiply each element of that row or column by its cofactor. The cofactor is the determinant of the submatrix obtained by removing the row and column of the selected element. After multiplying and summing up the products, you get the value of the determinant. However, this method can be computationally intensive for large matrices, so it is often more efficient to use calculators or software for higher order determinants.
Text Transcript from Video
a determinant is a single value that
represents a square matrix a third order
determinant is the determinant of a 3x3
matrix here on the Left we have a 3 by 3
matrix it consists of the elements a b c
d e f g h i and this is how you find its
third order determinate the first score
in each column is multiplied by its
minor so for example here's how we get
the first part we multiply a by e FHI
next we multiply B by D F G I and third
we multiply C by D e G H that's kind of
complicated so let's actually try a real
example here we have a 3 by 3 matrix
with elements 2 4 1 3 4 1 5 1 3
let's find its third order determinate
here the third order determinant is
equal to 2 times the matrix 4 1 1 3
minus 4 times the matrix 3 1 5 3 plus 1
times the matrix 3 4 5 1 using what we
know about scalar multiplication this
simplifies to 2 times 12 minus 1 minus 4
times 9 minus 5 plus 1 times 3 minus 20
this further simplifies to 20 minus 16
minus 17 for a final answer of negative
11 so the determinant of our 3x3 matrix
our third order determinant is equal to
negative 11
represents a square matrix a third order
determinant is the determinant of a 3x3
matrix here on the Left we have a 3 by 3
matrix it consists of the elements a b c
d e f g h i and this is how you find its
third order determinate the first score
in each column is multiplied by its
minor so for example here's how we get
the first part we multiply a by e FHI
next we multiply B by D F G I and third
we multiply C by D e G H that's kind of
complicated so let's actually try a real
example here we have a 3 by 3 matrix
with elements 2 4 1 3 4 1 5 1 3
let's find its third order determinate
here the third order determinant is
equal to 2 times the matrix 4 1 1 3
minus 4 times the matrix 3 1 5 3 plus 1
times the matrix 3 4 5 1 using what we
know about scalar multiplication this
simplifies to 2 times 12 minus 1 minus 4
times 9 minus 5 plus 1 times 3 minus 20
this further simplifies to 20 minus 16
minus 17 for a final answer of negative
11 so the determinant of our 3x3 matrix
our third order determinant is equal to
negative 11
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