Compute the adjoint of the following matrix: \[\begin{bmatrix}2 & - 1 & 3 \\ 4 & 2 & 5 \\ 0 & 4 & - 1\end{bmatrix}\] Verify that (adj A) A = |A| I = A (adj A) for the above matrix.

\[ C = \begin{bmatrix}2 & - 1 & 3 \\ 4 & 2 & 5 \\ 0 & 4 & - 1\end{bmatrix}\]Now, \[ C_{11} = \begin{vmatrix}2 & 5 \\ 4 & - 1\end{vmatrix} = - 22, C_{12} = - \begin{vmatrix}4 & 5 \\ 0 & - 1\end{vmatrix} = 4\text{ and }C_{13} = \begin{vmatrix}4 & 2 \\ 0 & 4\end{vmatrix} = 16\]\[ C_{21} = - \begin{vmatrix}- 1 & 3 \\ 4 & - 1\end{vmatrix} = 11, C_{22} = \begin{vmatrix}2 & 3 \\ 0 & - 1\end{vmatrix} = - 2\text{ and }C_{23} = - \begin{vmatrix}2 & - 1 \\ 0 & 4\end{vmatrix} = - 8\]\[ C_{31} = \begin{vmatrix}- 1 & 3 \\ 2 & 5\end{vmatrix} = - 11, C_{32} = - \begin{vmatrix}2 & 3 \\ 4 & 5\end{vmatrix} = 2\text{ and }C_{33} = \begin{vmatrix}2 & - 1 \\ 4 & 2\end{vmatrix} = 8\]\[ \therefore adjC = \begin{bmatrix}- 22 & 4 & 16 \\ 11 & - 2 & - 8 \\ - 11 & 2 & 8\end{bmatrix}^T = \begin{bmatrix}- 22 & 11 & - 11 \\ 4 & - 2 & 2 \\ 16 & - 8 & 8\end{bmatrix}\]\[(adjC)C = \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\]\[\text{ and }\left| C \right| = 0\]\[ \therefore \left| C \right|I = \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\]\[\text{ and }C\left( adjC \right) = \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\]\[\text{ Thus, }(adjA)A = \left| A \right|I = A(adjA)\]

Compute the Adjoint of the Following Matrix: ⎡ ⎢ ⎣ 2 − 1 3 4 2 5 0 4 − 1 ⎤ ⎥ ⎦ Verify that (Adj A) a = |A| I = a (Adj A) for the Above Matrix. - Mathematics

प्रश्न

Compute the adjoint of the following matrix:

[\begin{matrix} 2 & - 1 & 3 \\ 4 & 2 & 5 \\ 0 & 4 & - 1 \end{matrix}]

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.

उत्तर

$C = [\begin{matrix} 2 & - 1 & 3 \\ 4 & 2 & 5 \\ 0 & 4 & - 1 \end{matrix}]$
Now,
$C_{11} = | \begin{matrix} 2 & 5 \\ 4 & - 1 \end{matrix} | = - 22, C_{12} = - | \begin{matrix} 4 & 5 \\ 0 & - 1 \end{matrix} | = 4 and C_{13} = | \begin{matrix} 4 & 2 \\ 0 & 4 \end{matrix} | = 16$
$C_{21} = - | \begin{matrix} - 1 & 3 \\ 4 & - 1 \end{matrix} | = 11, C_{22} = | \begin{matrix} 2 & 3 \\ 0 & - 1 \end{matrix} | = - 2 and C_{23} = - | \begin{matrix} 2 & - 1 \\ 0 & 4 \end{matrix} | = - 8$
$C_{31} = | \begin{matrix} - 1 & 3 \\ 2 & 5 \end{matrix} | = - 11, C_{32} = - | \begin{matrix} 2 & 3 \\ 4 & 5 \end{matrix} | = 2 and C_{33} = | \begin{matrix} 2 & - 1 \\ 4 & 2 \end{matrix} | = 8$
$∴ a d j C = {[\begin{matrix} - 22 & 4 & 16 \\ 11 & - 2 & - 8 \\ - 11 & 2 & 8 \end{matrix}]}^{T} = [\begin{matrix} - 22 & 11 & - 11 \\ 4 & - 2 & 2 \\ 16 & - 8 & 8 \end{matrix}]$
$(a d j C) C = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$
$and | C | = 0$
$∴ | C | I = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$
$and C (a d j C) = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$
$Thus, (a d j A) A = | A | I = A (a d j A)$

shaalaa.com

Inverse of Matrix - Inverse of a Square Matrix by the Adjoint Method

या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?

Q 2.3 Q 2.2 Q 2.4

पाठ 7: Adjoint and Inverse of a Matrix - Exercise 7.1 [पृष्ठ २२]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12

पाठ 7 Adjoint and Inverse of a Matrix
Exercise 7.1 | Q 2.3 | पृष्ठ २२

व्हिडिओ ट्यूटोरियलVIEW ALL [3]

view
व्हिडिओ ट्यूटोरियल For All Subjects
Inverse of Matrix - Inverse of a Square Matrix by the Adjoint Method

video tutorial00:25:43
Inverse of Matrix - Inverse of a Square Matrix by the Adjoint Method

video tutorial00:21:40
Inverse of Matrix - Inverse of a Square Matrix by the Adjoint Method

video tutorial00:27:31

संबंधित प्रश्‍न

Find the adjoint of the matrices.

$[\begin{matrix} 1 & - 1 & 2 \\ 2 & 3 & 5 \\ - 2 & 0 & 1 \end{matrix}]$

Find the inverse of the matrices (if it exists).

$[\begin{matrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{matrix}]$

Find the inverse of the matrices (if it exists).

$[\begin{matrix} 1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4 \end{matrix}]$

For the matrix A = $[\begin{matrix} 3 & 2 \\ 1 & 1 \end{matrix}]$ find the numbers a and b such that A² + aA + bI = O.

For the matrix A = $[\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & - 3 \\ 2 & - 1 & 3 \end{matrix}]$ show that A³ − 6A² + 5A + 11 I = O. Hence, find A^−1.

Find the adjoint of the following matrix:
$[\begin{matrix} a & b \\ c & d \end{matrix}]$

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.

Find the adjoint of the following matrix:
$[\begin{matrix} \cos α & \sin α \\ \sin α & \cos α \end{matrix}]$

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.

Find the inverse of the following matrix.

[\begin{matrix} 1 & 0 & 0 \\ 0 & \cos α & \sin α \\ 0 & \sin α & - \cos α \end{matrix}]

Given $A = [\begin{matrix} 2 & - 3 \\ - 4 & 7 \end{matrix}]$ , compute A⁻¹ and show that $2 A^{- 1} = 9 I - A .$

Find the inverse of the matrix $A = [\begin{matrix} a & b \\ c & \frac{1 + b c}{a} \end{matrix}]$ and show that $a A^{- 1} = (a^{2} + b c + 1) I - a A .$

A = \frac{1}{9} [\begin{matrix} - 8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & - 8 & 4 \end{matrix}]

,
prove that

A^{- 1} = A^{3}

Solve the matrix equation $[\begin{matrix} 5 & 4 \\ 1 & 1 \end{matrix}] X = [\begin{matrix} 1 & - 2 \\ 1 & 3 \end{matrix}]$ , where X is a 2 × 2 matrix.

Find the matrix X for which

[\begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix}] X [\begin{matrix} - 1 & 1 \\ - 2 & 1 \end{matrix}] = [\begin{matrix} 2 & - 1 \\ 0 & 4 \end{matrix}]

If $A = [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}]$ , find $A^{- 1}$ and prove that $A^{2} - 4 A - 5 I = O$

If $A = [\begin{matrix} 1 & - 2 & 3 \\ 0 & - 1 & 4 \\ - 2 & 2 & 1 \end{matrix}], find {(A^{T})}^{- 1} .$

Find the adjoint of the matrix $A = [\begin{matrix} - 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1 \end{matrix}]$ and hence show that $A (a d j A) = | A | I_{3}$ .

Find the inverse by using elementary row transformations:

$[\begin{matrix} 1 & 2 & 0 \\ 2 & 3 & - 1 \\ 1 & - 1 & 3 \end{matrix}]$

If adj $A = [\begin{matrix} 2 & 3 \\ 4 & - 1 \end{matrix}] and adj B = [\begin{matrix} 1 & - 2 \\ - 3 & 1 \end{matrix}]$

If $A = [\begin{matrix} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{matrix}] and A (a d j A =) [\begin{matrix} k & 0 \\ 0 & k \end{matrix}]$ , then find the value of k.

If A is an invertible matrix such that |A⁻¹| = 2, find the value of |A|.

If A is an invertible matrix, then which of the following is not true ?

If A is an invertible matrix of order 3, then which of the following is not true ?

The matrix $[\begin{matrix} 5 & 10 & 3 \\ - 2 & - 4 & 6 \\ - 1 & - 2 & b \end{matrix}]$ is a singular matrix, if the value of b is _____________ .

If $A = [\begin{matrix} 2 & 3 \\ 5 & - 2 \end{matrix}]$ be such that $A^{- 1} = k A$ , then k equals ___________ .

If $A = [\begin{matrix} 1 & 0 & 1 \\ 0 & 0 & 1 \\ a & b & 2 \end{matrix}], then aI + bA + 2 A^{2}$ equals ____________ .

If a matrix A is such that $3 A^{3} + 2 A^{2} + 5 A + I = 0, then A^{- 1}$ equal to _______________ .

${(aA)}^{- 1} = \frac{1}{a} A^{- 1}$ , where a is any real number and A is a square matrix.

|A^–1| ≠ |A|^–1, where A is non-singular matrix.

Find the value of x for which the matrix A $= [\begin{matrix} 3 - x & 2 & 2 \\ 2 & 4 - x & 1 \\ - 2 & - 4 & - 1 - x \end{matrix}]$ is singular.

If A = [a_ij] is a square matrix of order 2 such that a_ij = ${\begin{cases} 1, when i \neq j \\ 0, when i = j \end{cases},$ then A² is ______.

For matrix A = $[\begin{matrix} 2 & 5 \\ - 11 & 7 \end{matrix}]$ (adj A)' is equal to:

A and B are invertible matrices of the same order such that |(AB)^-1| = 8, If |A| = 2, then |B| is ____________.

If for a square matrix A, A² – A + I = 0, then A^–1 equals ______.

Given that A is a square matrix of order 3 and |A| = –2, then |adj(2A)| is equal to ______.

To raise money for an orphanage, students of three schools A, B and C organised an exhibition in their residential colony, where they sold paper bags, scrap books and pastel sheets made by using recycled paper. Student of school A sold 30 paper bags, 20 scrap books and 10 pastel sheets and raised ₹ 410. Student of school B sold 20 paper bags, 10 scrap books and 20 pastel sheets and raised ₹ 290. Student of school C sold 20 paper bags, 20 scrap books and 20 pastel sheets and raised ₹ 440.

Answer the following question:

Translate the problem into a system of equations.
Solve the system of equation by using matrix method.
Hence, find the cost of one paper bag, one scrap book and one pastel sheet.

Compute the Adjoint of the Following Matrix: ⎡ ⎢ ⎣ 2 − 1 3 4 2 5 0 4 − 1 ⎤ ⎥ ⎦ Verify that (Adj A) a = |A| I = a (Adj A) for the Above Matrix. - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तर

APPEARS IN

व्हिडिओ ट्यूटोरियलVIEW ALL [3]

संबंधित प्रश्‍न

Select a course

Compute the Adjoint of the Following Matrix: ⎡ ⎢ ⎣ 2 − 1 3 4 2 5 0 4 − 1 ⎤ ⎥ ⎦ Verify that (Adj A) a = |A| I = a (Adj A) for the Above Matrix. - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तरShow Solution

APPEARS IN

व्हिडिओ ट्यूटोरियलVIEW ALL [3]

संबंधित प्रश्‍न

Select a course

उत्तर