###### Advertisements

###### Advertisements

Find the inverse of the following matrix by elementary row transformations if it exists. `A=[[1,2,-2],[0,-2,1],[-1,3,0]]`

###### Advertisements

#### Solution

`A=[[1,2,-2],[0,-2,1],[-1,3,0]]`

`therefore A=|[1,2,-2],[0,-2,1],[-1,3,0]|`

`=1|[-2,1],[3,0]|-2|[0,1],[-1,1]|-2|[0,-2],[-1,3]|`

`|A|=1(0-3)-2(0+1)-2(0-2)`

`=-3-2+4`

`=-1!=0`

`therefore A^(-1) " exist"`

We have

`A A^(-1)=I`

`[[1,2,-2],[0,-2,1],[-1,3,0]]A^(-1)=[[1,0,0],[0,1,0],[0,0,1]]`

`R_3->R_3+R_1`

`[[1,2,-2],[0,-2,1],[0,5,-2]]A^(-1)=[[1,0,0],[0,1,0],[1,0,1]]`

`R_3->R_3+2R_2`

`[[1,2,-2],[0,-2,1],[0,1,-0]]A^(-1)=[[1,0,0],[0,1,0],[1,2,1]]`

`R_2 harr R_3`

`[[1,2,-2],[0,1,0],[0,-2,1]]A^(-1)=[[1,0,0],[1,2,1],[0,1,0]]`

`R_1->R_1-2R_2 " " R3->R_3+2R_2`

`[[1,0,-2],[0,1,0],[0,0,1]]A^(-1)=[[-1,-4,-2],[1,2,1],[2,5,2]]`

`R_1->R_1+2R_3`

`[[1,0,0],[0,1,0],[0,0,1]]A^(-1)=[[3,6,2],[1,2,1],[2,5,2]]`

`A^(-1)=[[3,6,2],[1,2,1],[2,5,2]]`

#### APPEARS IN

#### RELATED QUESTIONS

The sum of three numbers is 6. If we multiply the third number by 3 and add it to the second number we get 11. By adding first and third numbers we get a number, which is double than the second number. Use this information and find a system of linear equations. Find these three numbers using matrices.

**Find the inverse of the following matrix by elementary row transformations if it exists.**`A = [(1, 2, -2), (0, -2, 1), (-1, 3, 0)]`

If A = `[(1, 3), (3, 1)]`, Show that A^{2} - 2A is a scalar matrix.

Find the co-factor of the element of the following matrix.

`[(1,-1,2),(-2,3,5),(-2,0,-1)]`

Find the matrix of the co-factor for the following matrix.

`[(1, 0, 2),(-2, 1, 3),(0, 3, -5)]`

Find the adjoint of the following matrix.

`[(2,-3),(3,5)]`

Find the adjoint of the following matrix.

`[(1, -1, 2),(-2, 3, 5),(-2, 0, -1)]`

If A = `[(1,-1,2),(3,0,-2),(1,0,3)]` verify that A (adj A) = (adj A) A = | A | I

Find the inverse of the following matrix by the adjoint method.

`[(-1,5),(-3,2)]`

Find the inverse of the following matrix.

`[(2, -3),(-1, 2)]`

Find the inverse of the following matrix.

`[(2,0,-1),(5,1,0),(0,1,3)]`

**Find the inverses of the following matrices by the adjoint method:**

`[(1,2,3),(0,2,4),(0,0,5)]`

**Find the inverse of the following matrix (if they exist):**

`((1,-1),(2,3))`

**Find the inverse of the following matrix (if they exist):**

`[(2,-3,3),(2,2,3),(3,-2,2)]`

**Find the inverse of the following matrix (if they exist):**

`[(2,0,-1),(5,1,0),(0,1,3)]`

Find the inverse of `[(1,2,3),(1,1,5),(2,4,7)]` by the adjoint method.

**Choose the correct answer from the given alternatives in the following question:**

If A = `[(lambda,1),(-1, -lambda)]`, and A^{-1} does not exist if λ = _______

**Choose the correct answer from the given alternatives in the following question:**

If A = `[("cos"alpha, - "sin"alpha,0),("sin"alpha,"cos"alpha,0),(0,0,1)]` where α ∈ R, then [F(α)]^{-1} is

**Choose the correct answer from the given alternatives in the following question:**

The inverse of A = `[(0,1,0),(1,0,0),(0,0,1)]` is

**Choose the correct answer from the given alternatives in the following question:**

The inverse of a symmetric matrix is

Find the inverse of the following matrices by the adjoint method `[(3, -1),(2, -1)]`.

Find the inverse of the following matrices by the adjoint method `[(2, -2),(4, 5)]`.

Find the inverse of the following matrices by the adjoint method `[(1, 2, 3),(0, 2, 4),(0, 0, 5)]`.

**Find the inverse of the following matrices by transformation method:**

`[(2, 0, −1),(5, 1, 0),(0, 1, 3)]`

Find the inverse of A = `[(1, 0, 1),(0, 2, 3),(1, 2, 1)]` by elementary column transformations.

Find matrix X, if AX = B, where A = `[(1, 2, 3),(-1, 1, 2),(1, 2, 4)] "and B" = [(1),(2),(3)]`.

Adjoint of `[(2, -3),(4, -6)]` is _______

**Choose the correct alternative.**

If A^{2} + mA + nI = O and n ≠ 0, |A| ≠ 0, then A^{–1} = _______

**Choose the correct alternative.**

If A is a 2 x 2 matrix such that A(adj. A) = `[(5, 0),(0, 5)]`, then |A| = _______

If A is a no singular matrix, then det (A^{–1}) = _______

If A = `[(1, 2),(-3, -1)], "B" = [(-1, 0),(1, 5)]`, then AB =

**Fill in the blank :**

If A = [a_{ij}]_{2x3} and B = [b_{ij}]_{mx1} and AB is defined, then m = _______

**Fill in the blank :**

If a_{1}x + b_{1}y = c_{1} and a_{2}x + b_{2}y = c_{2}, then matrix form is `[(......, ......),(......, ......)] = [(x),(y)] = [(......),(......)]`

**State whether the following is True or False :**

A(adj. A) = |A| I, where I is the unit matrix.

Solve the following :

If A = `[(2, -3),(3, -2),(-1, 4)],"B" = [(-3, 4, 1),(2, -1, -3)]`, verify (3A – 5B^{T})^{T} = 3A^{T} – 5B.

**Find inverse of the following matrices (if they exist) by elementary transformations :**

`[(1, -1),(2, 3)]`

**Find inverse of the following matrices (if they exist) by elementary transformations :**

`[(2, 1),(7, 4)]`

**Find inverse of the following matrices (if they exist) by elementary transformations :**

`[(2, -3, 3),(2, 2, 3),(3, -2, 2)]`

`cos theta [(cos theta, sin theta),(-sin theta, cos theta)] + sin theta [(sin theta, - cos theta),(cos theta, sin theta)]` = ______

If A = `[(4, 5),(2, 5)]`, then |(2A)^{−1}| = ______

If `[(x - y - z),(-y + z),(z)] = [(0),(5),(3)]`, then the value of x, y and z are respectively ______

If A = `[("a", "b"),("c", "d")]` then find the value of |A|^{−1}

If A = `[(1, 2),(3, -2),(-1, 0)]` and B = `[(1, 3, 2),(4, -1, 3)]` then find the order of AB

A + I = `[(3, -2),(4, 1)]` then find the value of (A + I)(A − I)

If A = `[(0, 3, 3),(-3, 0, -4),(-3, 4, 0)]` and B = `[(x),(y),(z)]`, find the matrix B'(AB)

If A is invertible matrix of order 3 and |A| = 5, then find |adj A|

If A = `[(2, 4),(1, 3)]` and B = `[(1, 1),(0, 1)]` then find (A^{−1} B^{−1})

Find A^{–1} using adjoint method, where A = `[(cos theta, sin theta),(-sin theta, cos theta)]`

Find the adjoint of matrix A = `[(6, 5),(3, 4)]`

Find the adjoint of matrix A = `[(2, 0, -1),(3, 1, 2),(-1, 1, 2)]`

If A = `[(1, 0, 0),(3, 3, 0),(5, 2, -1)]`, find A^{−1} by the adjoint method

**Choose the correct alternative:**

If A is a non singular matrix of order 3, then |adj (A)| = ______

**State whether the following statement is True or False:**

Inverse of `[(2, 0),(0, 3)]` is `[(1/2, 0),(0, 1/3)]`

If A = [a_{ij}]_{2×2}, where a_{ij} = i – j, then A = ______

The value of Cofactor of element a_{21} in matrix A = `[(1, 2),(5, -8)]` is ______

The value of Minor of element b_{22} in matrix B = `[(2, -2),(4, 5)]` is ______

Find the inverse of matrix B = `[(3,1, 5),(2, 7, 8),(1, 2, 5)]` by using adjoint method

Complete the following activity to verify A. adj (A) = det (A) I.

Given A = `[(2, 0, -1),(5, 1, 0),(0, 1, 3)]` then

|A| = 2(____) – 0(____) + ( ) (____)

= 6 – 0 – 5

= ______ ≠ 0

Cofactors of all elements of matrix A are

A_{11} = `(-1)^2 |("( )", "( )"),("( )", "( )")|` = (______),

A_{12} = `(-1)^3 |(5, "( )"),("( )", 3)|` = – 15,

A_{13} = `(-1)^4 |(5, "( )"),("( )", 1)|` = 5,

A_{21} = _______, A_{22} = _______, A_{23 }= _______,

A_{31} = `(-1)^4 |("( )", "( )"),("( )", "( )")|` = (______),

A_{32} = `(-1)^5 |(2, "( )"),("( )", 0)|` = ( ),

A_{33} = `(-1)^6 |(2, "( )"),("( )", 1)|` = 2,.

Cofactors of matrix A = `[(3, "____", "____"),("____", "____",-2),(1, "____", "____")]`

adj (A) = `[("____", "____", "____"),("____", "____","____"),("____","____","____")]`

A.adj (A) = `[(2, 0, -1),(5, 1, 0),(0, 1, 3)] [("( )", -1, 1), (-15, "( )", -5),("( )", -2, "( )")] = [(1, 0, "( )"),("( )", "( )", "( )"),(0, "( )", "( )")]` = |A|I

Complete the following activity to find inverse of matrix using elementary column transformations and hence verify.

`[(2, 0, -1),(5, 1, 0),(0, 1, 3)]` B^{−1} = `[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`

C_{1} → C_{1} + C_{3}

`[("( )", 0, -1),("( )", 1, 0),("( )", 1, 3)]` B^{−1} = `[("( )", 0, 0),("( )", 1, 0),("( )", 0, 1)]`

C_{3} → C_{3} + C_{1}

`[(1, 0, 0),("( )", 1, "( )"),(3, 1, "( )")]` B^{−1} = `[(1, 0, "( )"),(0, 1, 0),("( )", 0, "( )")]`

C_{1} → C_{1} – 5C_{2}, C_{3} → C_{3} – 5C_{2}

`[(1, "( )", 0),(0, 1, 0),("( )", 1, "( )")]` B^{−1} = `[(1, 0, "( )"),("( )", 1, -5),(1, "( )", 2)]`

C_{1} → C_{1} – 2C_{3}, C_{2} → C_{2} – C_{3 }

`[(1, 0, 0),(0, 1, 0),(0, 0, 1)]` B^{−1} = `[(3, -1, "( )"),("( )", 6, -5),(5, "( )", "( )")]`

B^{−1} = `[("( )", "( )", "( )"),("( )", "( )", "( )"),("( )", "( )", "( )")]`

`[(2, "( )", -1),("( )", 1, 0),(0, 1, "( )")] [(3, "( )", "( )"),("( )", 6, "( )"),("( )", -2, "( )")] = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]`

If A = `[(1,3,3),(1,4,3),(1,3,4)]` then verify that A(adj A) = |A| I and also find A^{-1}.

If A = `[(3,7),(2,5)]` and B = `[(6,8),(7,9)]`, then verify that (AB)^{-1} = B^{-1}A^{-1}

**Solve by matrix inversion method:**

x – y + 2z = 3; 2x + z = 1; 3x + 2y + z = 4

A sales person Ravi has the following record of sales for the month of January, February and March 2009 for three products A, B and C. He has been paid a commission at fixed rate per unit but at varying rates for products A, B and C.

Months |
Sales in units |
Commission |
||

A |
B |
C |
||

January | 9 | 10 | 2 | 800 |

February | 15 | 5 | 4 | 900 |

March | 6 | 10 | 3 | 850 |

Find the rate of commission payable on A, B and C per unit sold using matrix inversion method.

The prices of three commodities A, B, and C are ₹ x, y, and z per unit respectively. P purchases 4 units of C and sells 3 units of A and 5 units of B. Q purchases 3 units of B and sells 2 units of A and 1 unit of C. R purchases 1 unit of A and sells 4 units of B and 6 units of C. In the process P, Q and R earn ₹ 6,000, ₹ 5,000 and ₹ 13,000 respectively. By using the matrix inversion method, find the prices per unit of A, B, and C.

The inverse matrix of `((4/5,(-5)/12),((-2)/5,1/2))` is

Which of the following matrix has no inverse

If A and B non-singular matrix then, which of the following is incorrect?

If A is an invertible matrix of order 2 then det (A^{-1}) be equal

If A = `|(3,-1,1),(-15,6,-5),(5,-2,2)|` then, find the Inverse of A.

If A = `[(2,3),(1,2)]`, B = `[(1,0),(3,1)]`, then B^{-1}A^{-1} = ?

If A = `[(4,5),(2,1)]` and A^{2} - 5A - 6l = 0, then A^{-1} = ?

If [abc] ≠ 0, then `(["a" + "b b" + "c c" + "a"])/(["b c a"])` = ____________.

If ω is a complex cube root of unity and A = `[(ω,0,0),(0,ω^2,0),(0,0,1)]` then A^{-1} = ?

If A and Bare square matrices of order 3 such that |A| = 2, |B| = 4, then |A(adj B)| = ______.

If A = `[(2, -3), (3, 5)]`, then |Adj A| is equal to ______

If A^{2} - A + I = 0, then A^{-1} = ______.

If A = `[(1,-1,1),(2,1,-3),(1,1,1)]`, then the sum of the elements of A^{-1} is ______.

If A is a solution of x^{2} - 4x + 3 = 0 and `A=[[2,-1],[-1,2]],` then A^{-1} equals ______.

If A = `[(-i, 0),(0, i)]`, then A^{T}A is equal to

**Choose the correct option:**

If X, Y, Z are non zero real numbers, then the inverse of matrix A = `[(x, 0, 0),(0, y, 0),(0, 0, z)]`

If A, B are two square matries, such that AB = B, BA = A and n ∈ N then (A + B)^{n} =

If A = `[(2, 3),(a, 6)]` is a singular matrix, then a = ______.

Find the inverse of the matrix A by using adjoint method.

where A = `[(-3, -1, 1),(0, 0, 1),(-15, 6, -6)]`

If A = `[(cos α, sin α),(- sin α, cos α)]`, then the matrix A is ______.

If A = `[(1, 2),(3, 4)]` verify that A (adj A) = (adj A) A = |A| I

If A = `[(4, 3, 2),(-1, 2, 0)]`, B = `[(1, 2),(-1, 0),(1, -2)]`

Find (AB)^{–1} by adjoint method.

**Solution:**

AB = `[(4, 3, 2),(-1, 2, 0)] [(1, 2),(-1, 0),(1, -2)]`

AB = [ ]

|AB| = ** **`square`

M_{11} = –2 ∴ A_{11} = (–1)^{1+1} . (–2) = –2

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

M_{21} = 4 A_{21} = (–1)^{2+1} . (4) = –4

M_{22} = 3 A_{22} = (–1)^{2+2} . (3) = 3

Cofactor Matrix [Aij] = `[(-2, 3),(-4, 3)]`

adj (A) = [ ]

A^{–1} = `1/|A| . adj(A)`

A^{–1} = `square`

if `A = [(2,-1,1),(-1,2,-1),(1,-1,2)]` then find A^{−1} by the adjoint method.

If A = `[(2, 3),(4, 5)]`, show that A^{2} – 7A – 2I = 0

If A = `[(1, 2, 4),(4, 3, -2),(1, 0, -3)]`. Show that A^{–1} exists and find A^{–1} using column transformation.

If A = `[(3, 1),(-1, 2)]`, show that A^{2} – 5A + 7I = 0