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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=then determinant of A is written as

**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):

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.**Example:**

|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):

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!

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”):

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:**

|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,

**As a formula:**

Notice the +−+− pattern (+a… −b… +c… −d…).

**Solved Examples For You****Question 1:**

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:**

⇒ f(x) = x^{2} − 2λ^{2}

Therefore, f(λx) − f(x) = (λx)^{2} − 2λ^{2} − (x^{2}−2λ^{2})

=λ^{2}x^{2}−2λ^{2}−x^{2}+2λ^{2}

=x^{2}(λ^{2}−1)

So, option D is correct.

**Question 2:**

then (a^{2} + b^{2} – c^{2}) |A| =

1. abc

2. a + b + c

3. a3 + b3 + c3

4. 0**Solution:** |A| = 0 × (a^{2}) −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 a_{ij} of a determinant is the determinant obtained by deleting its i^{th} row and j^{th} column in which element a_{ij} lies. Minor of an element a_{ij} is denoted by M_{ij}.

**Cofactor of a Determinant**

The cofactor is defined as the signed minor. Cofactor of an element a_{ij}, denoted by A_{ij} is defined by A = (–1)^{i+j} M, where M is minor of a_{ij}.**Note**

We note that if the sum i+j is even, then A_{ij} = M_{ij}, and that if the sum is odd, then A_{ij} = −M_{ij}.

Hence, the only difference between the related minor entries and cofactors may be a sign change or nothing at all.

Whether or A_{ij} = M_{ij} or A_{ij} = −M_{ij}

has a pattern for square matrices as illustrated:**For example** C_{12} = −M_{12}. Of course, if you forget, you can always use the formula C_{ij} = (−1)^{i+j} M_{ij},

Here, C_{12}=(−1)^{1+2} M_{ij }= (−1)^{3} M_{ij} = −M_{ij}

**Example: **Find the minors and cofactors of all the elements of the determinant**Solution:** Minor of the element aij is Mij.

Here a_{11} = 1. So M_{11} = Minor of a_{11} = 3

M_{12} = Minor of the element a_{12} = 4

M_{21} = Minor of the element a_{21} = –2

M_{22} = Minor of the element a_{22} = 1

Now, cofactor of aij is Aij. So,

A_{11} = (–1)^{1+1}, M_{11} = (–1)^{2} (3) = 3

A_{12} = (–1)^{1+2}, M_{12 }= (–1)^{3} (4) = –4

A_{21} = (–1)^{2+1}, M_{21} = (–1)^{3} (–2) = 2

A_{22} = (–1)^{2+2}, M_{22} = (–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 = C^{T} where C is the cofactor matrix of A.

Also |Adj A|=|A|^{n−1}

Now |C^{T}| = |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

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

Cofactor of -4 is (−1)^{1+2}(42) = −42

Minor of 9 is

Cofactor of 9 is (−1)^{3+3} .(3) = 3. Therefore, the answer is option B

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