SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC chapter 1.2 - Matrics [Latest edition]

The adjoint matrix of `[(3, -3, 4),(2, -3, 4),(0, -1, 1)]` is ______.

A = `[(cos alpha, - sin alpha,  0),(sin alpha, cos alpha,  0),(0, 0, 1)]`, then A⁻¹ is

The solution (x, y, z) of the equation `[(1, 0, 1),(-1, 1, 0),(0, -1, 1)] [(x),(y),(z)] = [(1),(1),(2)]` is (x, y, z) =

If ω is a complex cube root of unity, then the matrix A = `[(1, ω^2, ω),(ω^2, ω, 1),(ω, 1, ω^2)]` is

If A = `[(4, -1),(-1, "k")]` such that A² − 6A + 7I = 0, then K = ______

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

If A = `[(0, 0, -1),(0, -1, 0),(-1, 0, 0)]`, then the only correct statement about the matrix A is ______

If A = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)]`, then A¹⁰ = ______

The element of second row and third column in the inverse of `[(1, 2, 1),(2, 1, 0),(-1, 0, 1)]` is ______.

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 ______

The value of x, y, z for the following system of equations x + y + z = 6, x − y+ 2z = 5, 2x + y − z = 1 are ______

If A = `[(3, 0, 0),(0, 3, 0),(0, 0, 3)]`, then |A| |adj A| = ______

System of equations x + y = 2, 2x + 2y = 3 has ______

If A = `[(1, -1, 1),(2, 1, -3),(1, 1, 1)]`, 10B = `[(4, 2,2),(-5, 0, ∞),(1, -2, 3)]` and B is the inverse of matrix A, then α = ______

If A = `[(1, 2, 3),(1, 1, 5),(2, 4, 7)]`, then find the value of a₃₁A₃₁ + a₃₂A₃₂ + a₃₃A₃₃.

For an invertible matrix A, if A . (adj A) = `[(10, 0),(0, 10)]`, then find the value of |A|.

If the inverse of the matrix `[(alpha, 14, -1),(2, 3, 1),(6, 2, 3)]` does not exists then find the value of α

If A = `[(2, 2),(-3, 2)]` and B = `[(0, -1),(1, 0)]`, then find the matrix (B⁻¹ A⁻¹)⁻¹.

A = `[(cos theta, - sin theta),(-sin theta, -cos theta)]` then find A⁻¹ 

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

If A = `[(3, 1),(5, 2)]`, and AB = BA = I, then find the matrix B

If A(α) = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)]` then prove that A²(α) = A(2α)

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 = `[(2, -1, 1),(-2, 3, -2),(-4, 4, -3)]` the find A² 

If A = `[(-2, 4),(-1, 2)]` then find A² 

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

If f(x) = x² − 2x − 3 then find f(A) when A = `[(1, 2),(2, 1)]`

If A = `[(-1),(2),(3)]`, B = `[(3, 1, -2)]`, find B'A'

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

If A = `[(6, 5),(5, 6)]` and B = `[(11, 0),(0, 11)]` then find A'B'

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

If A = `[(2, 0),(0, 1)]` and B = `[(1),(2)]`, then find the matrix X such that A⁻¹X = B.

Find the matrix X such that AX = I where A = `[(6, 17),(1, 3)]`

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

Find A⁻¹ using column transformations:

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

Find A⁻¹ using column transformations:

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

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

Transform `[(1, 2, 4),(3, -1, 5),(2, 4, 6)]` into an upper triangular matrix by using suitable row transformations

If A = `[(0, 4, 3),(1, -3, -3),(-1, 4, 4)]`, then find A² and hence find A⁻¹ 

If A = `[(0, 1),(2, 3),(1, -1)]` and B = `[(1, 2, 1),(2, 1, 0)]`, then find (AB)⁻¹ 

If A = `[(-4, -3, -3),(1, 0, 1),(4, 4, 3)]`, find adj (A).

Solve the following by inversion method 2x + y = 5, 3x + 5y = −3

If A = `[(1, 2, -1),(3, -2, 5)]`, apply R₁ ↔ R₂ and then C₁ → C₁ + 2C₃ on A

Three chairs and two tables cost ₹ 1850. Five chairs and three tables cost ₹2850. Find the cost of four chairs and one table by using matrices

If A = `[(4,5),(2,1)]`, show that `"A"^-1 = 1/6("A" - 5"I")`.

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

Find the matrix X such that `[(1, 2, 3),(2, 3, 2),(1, 2, 2)]` X = `[(2, 2, -5),(-2, -1, 4),(1, 0, -1)]`

Find the inverse of A = `[(sec theta, tan theta, 0),(tan theta, sec theta, 0),(0, 0, 1)]`

Transform `[(1, 2, 4),(3, -1, 5),(2, 4, 6)]` into an upper triangular matrix by using suitable row transformations

If A = `[(1, 0, 1),(0, 2, 3),(1, 2, 1)] "and B" = [(1, 2, 3),(1, 1, 5),(2, 4, 7)]`, then find a matrix X such that XA = B.

Find the inverse of A = `[(2, -3, 3),(2, 2, 3),(3, -2, 2)]` by using elementary row transformations.

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

Solve the following equations by using inversion method.

x + y + z = −1, x − y + z = 2 and x + y − z = 3

If three numbers are added, their sum is 2. If 2 times the second number is subtracted from the sum of first and third numbers, we get 8. If three times the first number is added to the sum of second and third numbers, we get 4. Find the numbers using matrices.

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

If A = `[("x",0,0),(0,"y",0),(0,0,"z")]` is a non-singular matrix, then find A⁻¹ by using elementary row transformations. Hence, find the inverse of `[(2,0,0),(0,1,0),(0,0,-1)]`

Find the inverse of A = `[("cos" theta, -"sin" theta, 0),("sin" theta, "cos" theta, 0),(0,0,1)]` by elementary row transformations.

If A = `[(1,1),(1,2)], "B" = [(4,1),(3,1)]` and C = `[(24,7),(31,9)]`, then find the matrix X such that AXB = C

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

If A = `[(2, 3),(1, 2)]`, B = `[(1, 0),(3, 1)]`, find AB and (AB)⁻¹ 

Solve the following system of equations by using inversion method

x + y = 1, y + z = `5/3`, z + x = `4/3`

The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is ₹ 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is ₹ 90. Whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is ₹ 70. Find the cost of each item per dozen by using matrices