Key Points
Key Points: Homogeneous and Non – Homogeneous Equations
| Condition | Non-Homogeneous (B ≠ 0) | Homogeneous (B = 0) |
|---|---|---|
| |A| ≠ 0 | Consistent, unique solution (X = A⁻¹B) | Consistent, only a trivial solution |
| |A| = 0 | If adj(A)·B ≠ 0 → Inconsistent (no solution) | Consistent |
| |A| = 0 | If adj(A)·B = 0 → Infinite solutions | Infinite (non-trivial) solutions |
| Equations < Variables | May have infinite solutions | A non-trivial solution exists |
Important Questions [15]
- Using Encoding Matrix [ 1 1 0 1 ] ,Encode and Decode the Message "Mumbai"
- Show that the Following Equations: -2x + Y + Z = A, X - 2y + Z = B, X + Y - 2z = C Have No Solutions Unless a +B + C = 0 in Which Case They Have Infinitely Many Solutions. Fin
- Using the Matrix a = [ − 1 2 − 1 1 ] Decode the Message of Matrix C= [ 4 11 12 − 2 − 4 4 9 − 2 ]
- P Investigate for What Values of 𝝁 "𝒂𝒏𝒅" 𝝀 the Equations : 2 X + 3 Y + 5 Z = 9 7 X + 3 Y − 2 Z = 8 2 X + 3 Y + λ Z = μ Have (I) No Solution (Ii) Unique Solution (Iii) Infinite Value
- Obtain the Root of X 3 − X − 1 = 0 by Newton Raphson Method` (Upto Three Decimal Places).
- Reduce Matrix to Paq Normal Form and Find 2 Non-singular Matrices P and Q. ⎡ ⎢ ⎣ 1 2 − 1 2 2 5 .2 3 1 2 1 2 ⎤ ⎥ ⎦
- Reduce the Following Matrix to Its Normal Form and Hence Find Its Rank.
- Find non singular matrices P & Q such that PAQ is in normal form where A ⎡ ⎢ ⎣ 2 − 2 3 3 − 1 2 1 2 − 1 ⎤ ⎥ ⎦
- Find Non Singular Matrices P and Q Such that a = ⎡ ⎢ ⎣ 1 2 3 2 2 3 5 1 1 3 4 5 ⎤ ⎥ ⎦
- If U = Sin − 1 ( X 1 3 + Y 1 3 X 1 2 − Y 1 2 ) , Prove that X 2 ∂ 2 U ∂ X 2 + 2 X Y ∂ 2 U ∂ X ∂ Y + Y 2 ∂ 2 U ∂ Y 2 = Tan U 144 ( Tan 2 U + 13 )
- If U= Sin − 1 ( X + Y √ X + √ Y ) , Prove that I.Xu_X+Yu_Y=1/2 Tanu` Ii. X 2 U × + 2 X Y U X Y + Y 2 U Y Y = − Sin U . Cos 2 U 4 Cos 3 U
- Investigate for what values of 𝝁 𝒂𝒏𝒅 𝝀 the equation x+y+z=6; x+2y+3z=10; x+2y+𝜆z=𝝁 have (i)no solution, (ii) a unique solution, (iii) infinite no. of solution.
- Investigate for What Values of μ and λ the Equations X+Y+Z=6, X+2y+3z=10, X+2y+λZ=μ Has 1) No Solution 2) a Unique Solution 3) Infinite Number of Solutions.
- Using the Encoding Matrix [ 1 1 0 1 ] Encode and Decode the Messag I*Love*Mumbai.
- Using Encoding Matrix [ 1 1 0 1 ] Encode and Decode the Message “All is Well” .
