![Determinant of a Matrix of Order One
• Determinant of a matrix of order one
A=[a11]1x1 is
𝐴 = a11 = a11](https://image.slidesharecdn.com/ljny8d00r3sykcpu6dql-signature-a4078f78569ff597de72b4bfa668ea6779f2fb21b7c17a22d8c7003b362b53f6-poli-161102164355/85/Determinants-2-320.jpg)
![Note:
• Multiplying a determinant by k means
multiply elements of only one row (or one
column) by k.
• If A = [aij]3×3,then | k.A| = k3 |A|](https://image.slidesharecdn.com/ljny8d00r3sykcpu6dql-signature-a4078f78569ff597de72b4bfa668ea6779f2fb21b7c17a22d8c7003b362b53f6-poli-161102164355/85/Determinants-13-320.jpg)
![Adjoint of a matrix
• Adjoint of a matrix A = [aij]n × n is the transpose of the
matrix [Aij]n × n, where Aij is the cofactor of the element aij .
Denoted by adj A.
• If A =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
adj A = Transpose of
𝐴11 𝐴12 𝐴13
𝐴21 𝐴22 𝐴23
𝐴31 𝐴32 𝐴33
=
𝐴11 𝐴21 𝐴31
𝐴12 𝐴22 𝐴32
𝐴13 𝐴23 𝐴33](https://image.slidesharecdn.com/ljny8d00r3sykcpu6dql-signature-a4078f78569ff597de72b4bfa668ea6779f2fb21b7c17a22d8c7003b362b53f6-poli-161102164355/85/Determinants-22-320.jpg)
![Theorem 4 : A square matrix A is invertible if and only if A is
nonsingular matrix
Verification: Let A be an invertible matrix. Then there exists a matrix
B such that AB = In = BA
⇒ |AB| =| In |
⇒ |A| |B| = I [ ∵ |AB| =|A| |B|]
⇒ |A| ≠ 0
⇒ A is a non-singular matrix
Conversly, let A be a non-singular matrix of order n. Then |A| ≠ 0
A(adj A) = |A|In = (adj A) A (by thm 1)
⇒ A
1
|A|
adj A = In =
1
|A|
adj A A
⇒ A-1 =
1
|A|
adj A
Hence A is an invertible matrix.](https://image.slidesharecdn.com/ljny8d00r3sykcpu6dql-signature-a4078f78569ff597de72b4bfa668ea6779f2fb21b7c17a22d8c7003b362b53f6-poli-161102164355/85/Determinants-27-320.jpg)
Determinants
- 1. DETERMINANTS • A Determinant of a matrix represents a single number. • We obtain this value by multiplying and adding its elements in a special way.
- 2. Determinant of a Matrix of Order One • Determinant of a matrix of order one A=[a11]1x1 is 𝐴 = a11 = a11
- 3. Determinant of a Matrix of Order Two • Determinant of a Matrix A= 𝑎11 𝑎12 𝑎21 𝑎22 2x2 is 𝐴 = 𝑎11 𝑎12 𝑎21 𝑎22 = 𝑎11 𝑎22 - 𝑎12 𝑎21
- 4. Determinant of a Matrix of Order Three • Determinant of a Matrix A = 𝑎11 𝑎12 𝑎13 𝑎21 𝑎22 𝑎23 𝑎31 𝑎32 𝑎33 is (expanding along R1) • 𝐴 = 𝑎11 𝑎12 𝑎13 𝑎21 𝑎22 𝑎23 𝑎31 𝑎32 𝑎33 = (-1)1+1 𝑎11 𝑎22 𝑎23 𝑎32 𝑎33 - (-1)1+2 𝑎12 𝑎21 𝑎23 𝑎31 𝑎33 + (-1)1+3 𝑎13 𝑎21 𝑎22 𝑎31 𝑎32 = 𝑎11(𝑎22 𝑎33 - 𝑎32 𝑎23) - 𝑎12(𝑎21 𝑎33 - 𝑎31 𝑎23) + 𝑎13(𝑎21 𝑎32 - 𝑎31 𝑎22) = 𝑎11 𝑎22 𝑎33 - 𝑎11 𝑎32 𝑎23 - 𝑎12 𝑎21 𝑎33 + 𝑎12 𝑎31 𝑎23 + 𝑎13 𝑎21 𝑎32 - 𝑎13 𝑎31 𝑎22
- 6. Evaluate the Determinant ∆ = 1 2 4 2 • 1 2 4 2 = 1(2) – 2(4) = 2 – 8 = -6
- 7. Find the Values of 𝑥 for which 3 𝑥 𝑥 1 = 3 2 4 1 • 3 𝑥 𝑥 1 = 3 2 4 1 • ie 3 – 𝑥2 = 3 - 8 ⇒ 𝑥2 = 8 ⇒ 𝑥 = ±2 2
- 8. Evaluate the Determinant ∆ = 3 −4 5 1 1 −2 2 3 1 • 3 −4 5 1 1 −2 2 3 1 = 3 1 −2 3 1 - (-4) 1 −2 2 1 + 5 1 1 2 3 = 3(1 + 6) + 4(1 + 4) + 5(3 -2) = 3(7) + 4(5) + 5(1) = 21 + 20 + 5 = 46
- 9. Properties of Determinants Property 1:The value of the determinant remains unchanged if its rows and columns are interchanged. Note: det(A) = det(A’) Where A’ = transpose of A
- 10. Property 2: If any two rows (or columns) of a determinant are interchanged,then sign of determinant changes
- 11. Property 3: If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then value of determinant is zero • Verification: If we interchange the identical rows (or columns) of the deteminent ∆, then ∆ does not change. But by Property 2 ∆ has changed its sign. Then ∆ = - ∆ ie ∆ = 0
- 12. Property 4: If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.
- 13. Note: • Multiplying a determinant by k means multiply elements of only one row (or one column) by k. • If A = [aij]3×3,then | k.A| = k3 |A|
- 14. Property 5:If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.
- 15. Property 6: To each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then the value of determinant remains the same
- 16. Area of a triangle • Area of a triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is ∆ = 1 2 x1 y1 1 x2 y2 1 x3 y3 1
- 17. Minor • Minor of an element aij of a determinant of order n is the determinant of the square sub- matrix of order (n-1) obtained by leaving ith row and j th column. • Minor of aij is denoted by Mij.
- 18. Find the minors of 1,-2,4,2 in 1 −2 4 2 • M11 = Minor of a11 = Minor of 1 = 2 • M12 = Minor of a12 = Minor of -2 = 4 • M21 = Minor of a21 = Minor of 4 = -2 • M22 = Minor of a22 = Minor of 2 = 1
- 19. Find the minors of all elements in 3 −4 5 1 6 −2 2 3 0 • M11 = Minor of a11 = Minor of 3 = 6 −2 3 0 = (6x0) – (-2x3) = 0 + 6 = 6 • M12 = Minor of a12 = Minor of -4= 1 −2 2 0 = (1x0) – (-2x2) = 0 + 4 = 4 • M13 = Minor of a13 = Minor of 5 = 1 6 2 3 = (1x3) – (2x6) = 3 -12 = -9 • M21 = Minor of a21 = Minor of 1 = −4 5 3 0 = (-4x0) – (3x5) = 0 -15 = -15 • M22 = Minor of a22 = Minor of 6 = 3 5 2 0 = (3x0) – (2x5) = 0 - 10 = -10 • M23 = Minor of a23 = Minor of -2= 3 −4 2 3 = (3x3) – (-4x2) = 9+8 = 17 • M31 = Minor of a31 = Minor of 2 = −4 5 6 −2 = (-4x-2)-(6x5) =8-30 =-22 • M32 = Minor of a32 = Minor of 3 = 3 5 1 −2 = (3x-2) – (1x5) = -6-5 =-11 • M33 = Minor of a33 = Minor of 0 = 3 −4 1 6 = (3x6) – (1x-4) = 18 +4 =22
- 20. Co factor • Cofactor of an element aij , denoted by Aij is Aij = (–1)i + j Mij where Mij is minor of aij. Note: Aij = 𝑀𝑖𝑗 𝑖𝑓 𝑖 + 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛 −𝑀𝑖𝑗 𝑖𝑓 𝑖 + 𝑗 𝑖𝑠 𝑜𝑑𝑑 If elements of a row (or column) are multiplied with cofactors of any other row (or column), then their sum is zero.
- 21. Find the cofactors of 1,-2,4,2 in 1 −2 4 2 • M11 = Minor of a11 = Minor of 1 = 2 A11= Cofactor of a11 = (-1)1+1M11 =2 M12 = Minor of a12 = Minor of -2 = 4 A12= Cofactor of a12 = (-1)1+2M12 = -4 • M21 = Minor of a21 = Minor of 4 = -2 A21= Cofactor of a21 = (-1)2+1M21 = 2 • M22 = Minor of a22 = Minor of 2 = 1 A22= Cofactor of a22 = (-1)2+2M22 = 1
- 22. Adjoint of a matrix • Adjoint of a matrix A = [aij]n × n is the transpose of the matrix [Aij]n × n, where Aij is the cofactor of the element aij . Denoted by adj A. • If A = 𝑎11 𝑎12 𝑎13 𝑎21 𝑎22 𝑎23 𝑎31 𝑎32 𝑎33 adj A = Transpose of 𝐴11 𝐴12 𝐴13 𝐴21 𝐴22 𝐴23 𝐴31 𝐴32 𝐴33 = 𝐴11 𝐴21 𝐴31 𝐴12 𝐴22 𝐴32 𝐴13 𝐴23 𝐴33
- 23. Theorem 1 : If A be any square matrix of order n, then A(adj A) = |A|In = (adj A) A Verification: Let A = 𝑎11 𝑎12 𝑎13 𝑎21 𝑎22 𝑎23 𝑎31 𝑎32 𝑎33 then adj A = 𝐴11 𝐴21 𝐴31 𝐴12 𝐴22 𝐴32 𝐴13 𝐴23 𝐴33 Sum of the product of elements of a row (or a column) with corresponding cofactor is equal to lAl, otherwise zero. ∴ A (adj A) = lAl 0 0 0 lAl 0 0 0 lAl = lAl 1 0 0 0 1 0 0 0 1 = lAl I ..…. (i) Similarly (adj A) A = lAl I …… (ii) By (i) & (ii) A(adj A) = |A|I n = (adj A) A
- 24. SINGULAR & NON SINGULAR • Singular: A square matrix A is said to be singular if | A | = 0 • Non Singular A square matrix A is said to be non-singular if |A| ≠ 0.
- 25. Theorem 2: If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order. i.e. If |A|≠ 0 & |B|≠ 0 Then |AB|≠ 0 & |BA|≠ 0 Theorem 3: The determinant of the product of matrices is equal to the product of their respective determinants. i.e. |AB| =|A| |B|
- 26. Invertible Matrices • If A & B are Square Matrices such that AB = BA = I B is called inverse matrix of A B = A-1 A is said to be invertible
- 27. Theorem 4 : A square matrix A is invertible if and only if A is nonsingular matrix Verification: Let A be an invertible matrix. Then there exists a matrix B such that AB = In = BA ⇒ |AB| =| In | ⇒ |A| |B| = I [ ∵ |AB| =|A| |B|] ⇒ |A| ≠ 0 ⇒ A is a non-singular matrix Conversly, let A be a non-singular matrix of order n. Then |A| ≠ 0 A(adj A) = |A|In = (adj A) A (by thm 1) ⇒ A 1 |A| adj A = In = 1 |A| adj A A ⇒ A-1 = 1 |A| adj A Hence A is an invertible matrix.
- 28. Consistency of the System of Linear Equation Consistent system A system of equations is said to be consistent if its solution (one or more) exists. Inconsistent system A system of equations is said to be inconsistent if its solution does not exist.
- 29. Solution of system of linear equations using inverse of a matrix If a1x+b1y+c1z = d1 a2x+b2y+c2z = d2 a3x+b3y+c3z = d3 writing these equation as AX = B where A = 𝑎1 𝑏1 𝑐1 𝑎2 𝑏2 𝑐2 𝑎3 𝑏3 𝑐3 , X = 𝑥 𝑦 𝑧 and B = 𝑑1 𝑑2 𝑑3 Then X = A-1B (i) |A|≠ 0 , there exists unique solution. (ii) |A| = 0 and (adj A)B ≠ 0, then there exists no solution. (iii)|A| = 0 and (adj A)B = 0, then system may or may not be consistent.