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The Determinant of a Matrix
To every square matrix A = [aij] of order n, we can associate a number (real or complex) called determinant of the square matrix A, where a = (i, j)th element of A. This may be thought of as a function which associates each square matrix with a unique number (real or complex).
If M is the set of square matrices, K is the set of numbers (real or complex) and f : M → K is defined by f (A) = k, where A ∈ M and k ∈ K, then f (A) is called the determinant of A. It is also denoted by | A | or det A or Δ.
If A=Determinant of a Matrix | Algebra - Mathematicsthen determinant of A is written as Determinant of a Matrix | Algebra - Mathematics

For a 1×1 Matrix
Let A = [a] be the matrix of order 1, then determinant of A is defined to be equal to a.

For a 2×2 Matrix
For a 2×2 matrix (2 rows and 2 columns):
Determinant of a Matrix | Algebra - Mathematics
The determinant is: |A| = ad − bc or the determinant of A equals a × d minus b × c. It is easy to remember when you think of a cross, where blue is positive that goes diagonally from left to right and red is negative that goes diagonally from right to left.
Determinant of a Matrix | Algebra - Mathematics
Example:
Determinant of a Matrix | Algebra - Mathematics

|A| = 2 x 8 – 4 x 3
= 16 – 12
= 4

For a 3×3 Matrix
For a 3×3 matrix (3 rows and 3 columns):
Determinant of a Matrix | Algebra - Mathematics
The determinant is: |A| = a (ei − fh) − b (di − fg) + c (dh − eg). The determinant of A equals ‘a times e x i minus f x h minus b times d x i minus f x g plus c times d x h minus e x g’. It may look complicated, but if you carefully observe the pattern its really easy!
Determinant of a Matrix | Algebra - Mathematics
To work out the determinant of a matrix 3×3:
• Multiply ‘a’ by the determinant of the 2×2 matrix that is not in a’s row or column.
• Likewise for ‘b’ and for ‘c’
• Sum them up, but remember the minus in front of the b
As a formula (remember the vertical bars || mean “determinant of”):
Determinant of a Matrix | Algebra - Mathematics
The determinant of A equals ‘a’ times the determinant of e × i minus f × h minus ‘b’ times the determinant of d × i minus f × g plus ‘c’ times the determinant of d × h minus e × g.
Example:
Determinant of a Matrix | Algebra - Mathematics
|A|= 6×(−2×7 − 5×8) − 1×(4×7 − 5×2) + 1×(4×8 − (−2×2))
= 6×(−54) − 1×(18) + 1×(36)
= −306

For 4×4 Matrices and Higher
The pattern continues for the determinant of a matrix 4×4:
• plus a times the determinant of the matrix that is not in a’s row or column,
minus b times the determinant of the matrix that is not in b’s row or column,
plus c times the determinant of the matrix that is not in c’s row or column,
 minus d times the determinant of the matrix that is not in d’s row or column,
Determinant of a Matrix | Algebra - Mathematics

As a formula:
Determinant of a Matrix | Algebra - Mathematics
Notice the +−+− pattern (+a… −b… +c… −d…).

Solved Examples For You
Question 1:
Determinant of a Matrix | Algebra - Mathematics
then f(λx) – f(x) is equal to:
A. x (λ2 – 1)
B. 2λ (x2 – 1)
C. λ2(x2 – 1)
D. x2 (λ2 – 1)
Solution:
Determinant of a Matrix | Algebra - Mathematics
⇒ f(x) = x2 − 2λ2
Therefore, f(λx) − f(x) = (λx)2 − 2λ2 − (x2−2λ2)
2x2−2λ2−x2+2λ2
=x22−1)
So, option D is correct.

Question 2:
Determinant of a Matrix | Algebra - Mathematics
then (a2 + b2 – c2) |A| =
1. abc
2. a + b + c
3. a3 + b3 + c3
4. 0
Solution: |A| = 0 × (a2) −c(− ab) − b(ac) = 0 + abc – abc = 0

Minor of a Determinant
A minor is defined as a value computed from the determinant of a square matrix which is obtained after crossing out a row and a column corresponding to the element that is under consideration. Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and jth column in which element aij lies. Minor of an element aij is denoted by Mij.

Cofactor of a Determinant
The cofactor is defined as the signed minor. Cofactor of an element aij, denoted by Aij is defined by A = (–1)i+j M, where M is minor of aij.
Note
We note that if the sum i+j is even, then Aij = Mij, and that if the sum is odd, then Aij = −Mij.
Hence, the only difference between the related minor entries and cofactors may be a sign change or nothing at all.
Whether  or Aij = Mij or Aij = −Mij
has a pattern for square matrices as illustrated:
Determinant of a Matrix | Algebra - Mathematics
For example C12 = −M12. Of course, if you forget, you can always use the formula Cij = (−1)i+j Mij,
Here, C12=(−1)1+2 Mij = (−1)3 Mij = −Mij

Example: Find the minors and cofactors of all the elements of the determinant
Determinant of a Matrix | Algebra - Mathematics
Solution: Minor of the element aij is Mij.
Here a11 = 1. So M11 = Minor of a11 = 3
M12 = Minor of the element a12 = 4
M21 = Minor of the element a21 = –2
M22 = Minor of the element a22 = 1
Now, cofactor of aij is Aij. So,
A11 = (–1)1+1, M11 = (–1)2 (3) = 3
A12 = (–1)1+2, M12 = (–1)3 (4) = –4
A21 = (–1)2+1, M21 = (–1)3 (–2) = 2
A22 = (–1)2+2, M22 = (–1)4 (1) = 1

Solved Examples for You
Question 1: Let A=[aij]n×n be a square matirx and let cij be cofactor of aij in A. If C=[cij], then
1. |A| = |C|
2.|C| = |A|n-1
3. |C| = |A|n-2
4. none of these
Solution: We know that adjA = CT where C is the cofactor matrix of A.
Also |Adj A|=|A|n−1
Now |CT| = |Adj A|
=|A|n-1 where n it the order of the square matrix.

Question 2: The minors and cofactors of -4 and 9 in determinant
Determinant of a Matrix | Algebra - Mathematics
are respectively
A. 42, 42; 3, 3
B. 42, -42; 3, 3
C. 42, -42; 3, -3
D. 42, 3; 42, 3
Solution: Minor of -4 is
Determinant of a Matrix | Algebra - Mathematics
Cofactor of -4 is (−1)1+2(42) = −42
Minor of 9 is
Determinant of a Matrix | Algebra - Mathematics
Cofactor of 9 is (−1)3+3 .(3) = 3. Therefore, the answer is option B

The document Determinant of a Matrix | Algebra - Mathematics is a part of the Mathematics Course Algebra.
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FAQs on Determinant of a Matrix - Algebra - Mathematics

1. What is the determinant of a matrix?
Ans. The determinant of a matrix is a scalar value that can be calculated from the elements of the matrix. It provides important information about the matrix, such as whether the matrix is invertible or singular.
2. How is the determinant of a matrix calculated?
Ans. The determinant of a matrix can be calculated using various methods, such as finding the cofactor expansion along a row or column, or using properties of determinants such as row operations or triangular matrices.
3. What are the properties of determinants?
Ans. Determinants have several properties, including linearity, multiplicative properties, and properties related to row operations. These properties allow us to simplify calculations and manipulate matrices efficiently.
4. What does the determinant tell us about a matrix?
Ans. The determinant provides important information about a matrix. If the determinant is non-zero, the matrix is invertible, meaning it has an inverse matrix. If the determinant is zero, the matrix is singular, and it does not have an inverse.
5. How is the determinant used in solving systems of equations?
Ans. The determinant can be used to solve systems of linear equations by representing the coefficients of the variables in a matrix form. By calculating the determinant, we can determine whether the system has a unique solution, infinitely many solutions, or no solution at all.
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