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You can correlate a value (real or complex) called the determinant of the square matrix A with every square matrix A = [] of order n, where a = (i,j)th member of A. This can be thought of as a function that assigns a unique number to each square matrix (real or complex).
If M is a collection of square matrices, K is a set of real or complex integers, and f (A) = k will be defined as f: M → k, where A ∈ M and k ∈ K, then f (A) is known as the determinant of A. You can also write this as | A |, A, or ∆.
The determinant can be thought of as a function with a square matrix as its input and a real number as its output. We can name our matrix if n is the number of rows and columns in the matrix. The simplest square matrix is 1×1, which isn’t important because it only includes one integer.
The determinant of this matrix is calculated by following these rules. We’ll start at the upper left component and work our way down the first row. By ignoring a’s row and column, we multiply component a by the determinant of the “submatrix” created. This submatrix is the 1×1 matrix consisting of d in this example, and its determinant is just d. As a result, the determinant’s first term is an ad.
Next, we’ll look at the upper right component b, which is the second component of the first row. The determinant of the submatrix produced by ignoring b’s row and column, which is c, is multiplied by b. As a result, the determinant’s following term is bc. Simply subtract the first term ad from the second term bc to get the overall determinant.
It’s a mess. Let’s understand it with a simple example:
Determinant of matrix questions with solutions
Question 1: Find the determinants of the 3x 3 matrix?
2 2 1 | -3 0 4 | 1 -1 5 |
Solution: First, you need to find the correspondence between elements of the real problem and generic elements in the formula.
a b c | d e f | g h i |
2 2 1 | -3 0 4 | 1 -1 5 |
Now you can use the determinants formula:
Applying the formula in the equation:
= 2[ 0 – (-4)] + 3 [10 – (-1)] +1 [8-0]
= 2 (0+4) +3 (10 +1) + 1(8)
= 2(4) +3(11) + 8
= 8+33+8
= 49
Therefore the answer would be given as:
2 2 1 | -3 0 4 | 1 -1 5 |
=49
Question 2: Determine the determinants of the 3×3 matrix?
1 -3 2 | 3 -1 3 | 2 -3 1 |
Solution: First, you need to find the correspondence between elements of the real problem and generic elements in the formula.
a b c | d e f | g h i |
1 -3 2 | 3 -1 3 | 2 -3 1 |
Now you can use the determinants formula:
Applying the formula in the equation:
= 1[ -1 – (-9)] – 3 [-3 – (-6)] + 2 [-9 – (-2)]
= 1 (-1+9) -3 (-3 +6) + 2(-9 + 2)
= 1(8) -3(3) +2(-7)
= 8 -9-14
= -15
Therefore the determinants of the equation would be:
1 -3 2 | 3 -1 3 | 2 -3 1 |
= -15
Question 3: Determine the determinants of the 3×3 matrix?
2 6 1 | 3 5 4 | 1 2 7 |
Solution: First, you need to find the correspondence between elements of the real problem and generic elements in the formula.
2 6 1 | 3 5 4 | 1 2 7 |
|A| = 2(35-8) – 3(42-2) +1(24-5)
|A| = 2(27) – 3(40) + 1(19)
|A| = 54-120+19
|A| = 73 -120
|A| = -47
Question 4: Determine the determinants of the 3×3 matrix?
4 3 | 2 2 |
Solution: First, you need to find the correspondence between elements of the real problem and generic elements in the formula.
4 3 | 2 2 |
The determinant of matrix A is
(A) = |A| = 8 – 6
|A| = 2
Question 5: Determine the determinants of the 3×3 matrix?
P=
√3/2 -½ | ½ √3/2 |
A =
1 0 | 0 1 |
and Q = PAPT, then P (Q2005)PT equal to __?
Solution:
If Q = PAPT, PT Q = APT,
(as PPT = I) PT Q2005 P = A PT Q2004 P
= A2 PT Q2003 P
= A3 PT Q2002 P
= A2004 PT (QP)
= A2004 PT(PA) (Q = PART ⇒ QP = PA)
= A2005
A2005 =
1 0 | 2005 1 |
Conclusion
Therefore, from the above article, we determined the definition of the determinant of a matrix, learnt about the determinant of a positive definite matrix and determinant of a symmetric positive definite matrix. A determinant can be defined as a special number that can be obtained from the matrix. The matrix is square-shaped( with the same number of rows and columns).
