Question:

Find the inverse of the matrices , if it exists. $\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the matrix.

Determinant of the matrix = 1 × 7 - 3 × 2 = 5

Step 2: Since the determinant is not equal to 0, the inverse of the matrix exists.

Step 3: Calculate the adjoint of the matrix.

Adjoint of the matrix = $\left[\begin{array}{cc}7& -3\\ -2& 1\end{array}\right]$

Step 4: Divide the adjoint by the determinant of the matrix.

Inverse of the matrix = $\left[\begin{array}{cc}7& -3\\ -2& 1\end{array}\right]/5$

= $\left[\begin{array}{cc}7& \frac{-3}{5}\\ \frac{-2}{5}& 1\end{array}\right]$

Question:

Find the inverse of the matrices, if it exists. $\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the matrix.

The determinant of the matrix is $\left|\begin{array}{cc}2& 5\\ 1& 3\end{array}\right|=1-15=-13$

Step 2: Check if the determinant is not equal to zero.

The determinant is not equal to zero, so the inverse of the matrix exists.

Step 3: Calculate the adjoint of the matrix.

The adjoint of the matrix is $\left[\begin{array}{cc}3& -5\\ -1& 2\end{array}\right]$

Step 4: Divide the adjoint matrix by the determinant.

The inverse of the matrix is $\left[\begin{array}{cc}3& -5\\ -1& 2\end{array}\right]/-13]=[\begin{array}{cc}3/-13& -5/-13\\ -1/-13& 2/-13\end{array}]=[\begin{array}{cc}-3/13& 5/13\\ 1/13& -2/13\end{array}]$

Question:

Find the inverse of the matrices, if it exists. $\left[\begin{array}{cc}4& 5\\ 3& 4\end{array}\right]$

Answer:

Step 1: Find the determinant of the matrix.

Determinant of the matrix = 4 × 4 - 5 × 3 = 16 - 15 = 1

Step 2: Since the determinant is not 0, the inverse of the matrix exists.

Step 3: Calculate the adjoint of the matrix.

Adjoint of the matrix = $\left[\begin{array}{cc}4& -5\\ -3& 4\end{array}\right]$

Step 4: Divide the adjoint of the matrix by the determinant to get the inverse of the matrix.

Inverse of the matrix = $\left[\begin{array}{cc}4& -5\\ -3& 4\end{array}\right]/1$

= $\left[\begin{array}{cc}4& -5\\ -3& 4\end{array}\right]$

Question:

Find the inverse of the matrices , if it exists. $\left[\begin{array}{cc}2& -3\\ -1& 2\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the matrix.

Determinant = 2 × 2 - (-3) × (-1) = 7

Step 2: Calculate the inverse of the matrix, if it exists.

Inverse = (1/7) × $\left[\begin{array}{cc}2& -3\\ -1& 2\end{array}\right]$

= (1/7) × $\left[\begin{array}{cc}2& -3\\ -1& 2\end{array}\right]$

= (1/7) × $\left[\begin{array}{cc}2& -3\\ -1& 2\end{array}\right]$

= (1/7) × $\left[\begin{array}{cc}2& 3\\ 1& -2\end{array}\right]$

= $\left[\begin{array}{cc}\mathrm{2/7}& \mathrm{3/7}\\ \mathrm{1/7}& \mathrm{-2/7}\end{array}\right]$

Therefore, the inverse of the given matrix is $\left[\begin{array}{cc}\mathrm{2/7}& \mathrm{3/7}\\ \mathrm{1/7}& \mathrm{-2/7}\end{array}\right]$.

Question:

Find the inverse of the matrices, if it exists. $\left[\begin{array}{cc}3& -1\\ -4& 2\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the matrix.

$\left[\begin{array}{cc}3& -1\\ -4& 2\end{array}\right]$

Determinant = 32 - (-1)(-4) = 10

Step 2: Check if the determinant is non-zero.

The determinant is 10, which is non-zero. Therefore, the inverse of the matrix exists.

Step 3: Calculate the adjoint of the matrix.

$\left[\begin{array}{cc}3& -1\\ -4& 2\end{array}\right]$

Adjoint = $\left[\begin{array}{cc}2& 4\\ -1& 3\end{array}\right]$

Step 4: Divide the adjoint matrix by the determinant.

$\left[\begin{array}{cc}2& 4\\ -1& 3\end{array}\right]$/10

Inverse = $\left[\begin{array}{cc}0.2& 0.4\\ -0.1& 0.3\end{array}\right]$

Question:

Find the inverse of each of the matrices, if it exists. $\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the matrix.

The determinant of the matrix is 5.

Step 2: Calculate the inverse of the matrix.

The inverse of the matrix is $\left[\begin{array}{cc}3& -1\\ -2& 1\end{array}\right]$ divided by the determinant (5).

Therefore, the inverse of the matrix is $\left[\begin{array}{cc}3& -1\\ -2& 1\end{array}\right]$/5.

Question:

Find the inverse of each of the matrices , if it exists. $\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the matrix.

Determinant = 2 - 1 = 1

Step 2: Since the determinant is not zero, the matrix is invertible.

Step 3: Calculate the adjoint of the matrix.

$\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]$

Step 4: Multiply the adjoint by 1/determinant.

$\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right] ÷ 1]$

Step 5: The inverse of the matrix is

$\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]$

Question:

Matrices A and B will be inverse of each other only if A AB=BA B AB=0,BA=I C AB=BA=0 D AB=BA=I

Answer:

A) Matrices A and B will be inverse of each other only if A × B = B × A

B) A × B × A = B × A × B = 0

C) B × A × B = A × B × A = 0

D) A × B × A = B × A × B = I (where I is the identity matrix)

Question:

Find the inverse of the following of the matrices , if it exists. $\left[\begin{array}{ccc}2& -3& 3\\ 2& 2& 3\\ 3& -2& 2\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the matrix.

The determinant of the matrix is 12.

Step 2: If the determinant is not equal to 0, then the inverse of the matrix exists.

Since the determinant is 12, the inverse of the matrix exists.

Step 3: Use the formula to calculate the inverse of the matrix.

The inverse of the matrix is

$\left[\begin{array}{ccc}1& 1& -2\\ -1& 2& 1\\ 2& -1& 1\end{array}\right]$

Question:

Find the inverse of the matrices, if it exists. $\left[\begin{array}{cc}3& 10\\ 2& 7\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the matrix.

The determinant of the matrix is $\left[\begin{array}{cc}3& 10\\ 2& 7\end{array}\right]$ = 37 - 102 = 17 - 20 = -3

Step 2: Check if the determinant is non-zero.

Since the determinant is -3, it is non-zero, so the matrix is invertible.

Step 3: Calculate the adjoint of the matrix.

The adjoint of the matrix is $\left[\begin{array}{cc}7& -10\\ -2& 3\end{array}\right]$

Step 4: Divide each element of the adjoint matrix by the determinant.

The inverse of the matrix is $\left[\begin{array}{cc}\frac{7}{-3}& \frac{-10}{-3}\\ \frac{-2}{-3}& \frac{3}{-3}\end{array}\right]$

Therefore, the inverse of the matrix is $\left[\begin{array}{cc}-7& 10\\ 2& -3\end{array}\right]$.

Question:

Find the inverse of the matrices, if it exists. $\left[\begin{array}{cc}2& 1\\ 7& 4\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the matrix:

$\left|\begin{array}{cc}2& 1\\ 7& 4\end{array}\right|=2-7=\mathrm{-5}$

Step 2: Check to see if the determinant is non-zero. Since the determinant is -5, the inverse of the matrix exists.

Step 3: Find the adjoint of the matrix.

$\left[\begin{array}{cc}2& 1\\ 7& 4\end{array}\right]\odot \left[\begin{array}{cc}4& -7\\ -1& 2\end{array}\right]=\left[\begin{array}{cc}4& 7\\ \mathrm{-1}& 2\end{array}\right]$

Step 4: Divide each element of the adjoint matrix by the determinant.

$\left[\begin{array}{cc}4& 7\\ \mathrm{-1}& 2\end{array}\right]÷-5=\left[\begin{array}{cc}\mathrm{¼4}& \mathrm{¼7}\\ 1& \mathrm{¼2}\end{array}\right]$

Step 5: The inverse of the matrix is:

$\left[\begin{array}{cc}\mathrm{¼4}& \mathrm{¼7}\\ 1& \mathrm{¼2}\end{array}\right]$

Question:

Find the inverse of the matrix, if it exists. $\left[\begin{array}{ccc}2& 0& -1\\ 5& 1& 0\\ 0& 1& 3\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the matrix.

Determinant = 2 × 1 × 3 + 0 × 0 × (-1) + 5 × (-1) × 0 = 6

Step 2: Check if the determinant is not zero.

Since the determinant is not zero, the matrix is invertible.

Step 3: Find the adjoint of the matrix.

Adjoint = $\left[\begin{array}{ccc}1& 0& 5\\ 0& 1& -1\\ -3& 0& 2\end{array}\right]$

Step 4: Divide the adjoint by the determinant.

Inverse = $\left[\begin{array}{ccc}\mathrm{1/6}& 0& \mathrm{5/6}\\ 0& \mathrm{1/6}& \mathrm{-1/6}\\ \mathrm{-1/2}& 0& \mathrm{1/3}\end{array}\right]$

Therefore, the inverse of the given matrix is $\left[\begin{array}{ccc}\mathrm{1/6}& 0& \mathrm{5/6}\\ 0& \mathrm{1/6}& \mathrm{-1/6}\\ \mathrm{-1/2}& 0& \mathrm{1/3}\end{array}\right]$.

Question:

Find the inverse of the matrices, if it exists. $\left[\begin{array}{cc}2& -6\\ 1& -2\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the matrix.

Determinant = 2(-2) - (-6) = 4 + 6 = 10

Step 2: Calculate the adjoint of the matrix.

Adjoint = $\left[\begin{array}{cc}-2& 6\\ 1& 2\end{array}\right]$

Step 3: Divide the adjoint by the determinant.

Inverse = $\left[\begin{array}{cc}-2& 6\\ 1& 2\end{array}\right]/10$

Inverse = $\left[\begin{array}{cc}\mathrm{-2/10}& \mathrm{6/10}\\ \mathrm{1/10}& \mathrm{2/10}\end{array}\right]$

Question:

Find the inverse of the matrices, if it exists. $\left[\begin{array}{cc}3& 1\\ 5& 2\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the matrix.

$\left|\begin{array}{cc}3& 1\\ 5& 2\end{array}\right|=3-10=-7$

Step 2: Check if the determinant is non-zero.

The determinant is not zero, so the matrix is invertible.

Step 3: Calculate the adjugate of the matrix.

$\begin{array}{cc}2& -1\\ -5& 3\end{array}$

Step 4: Divide the adjugate by the determinant.

$\frac{\begin{array}{cc}2& -1\\ -5& 3\end{array}}{7}$

Step 5: Simplify the fraction.

$\begin{array}{cc}2/7& -1/7\\ -5/7& 3/7\end{array}$

Step 6: The inverse of the matrix is:

$\left[\begin{array}{cc}2/7& -1/7\\ -5/7& 3/7\end{array}\right]$

Question:

Find the inverse of the matrices , if it exists. $\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the matrix.

Determinant = 27 - 53 = 1

Step 2: Calculate the adjoint of the matrix.

$\left[\begin{array}{cc}7& -3\\ -5& 2\end{array}\right]$

Step 3: Divide each element of the adjoint matrix by the determinant.

$\left[\begin{array}{cc}7& -3\\ -5& 2\end{array}\right]$/1 =

$\left[\begin{array}{cc}7& -3\\ -5& 2\end{array}\right]$

Step 4: The inverse of the given matrix is:

$\left[\begin{array}{cc}7& -3\\ -5& 2\end{array}\right]$

Question:

Find the inverse of the matrices , if it exists. $\left[\begin{array}{cc}6& -3\\ -2& 1\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the given matrix.

Determinant = 6 × 1 - (-3) × (-2) = 12 + 6 = 18

Step 2: Calculate the cofactors of each element in the matrix.

Cofactors of 6 = 1 Cofactors of -3 = 2 Cofactors of -2 = -3 Cofactors of 1 = 6

Step 3: Calculate the adjoint of the matrix by taking the transpose of the matrix of cofactors.

Adjoint = [1 2] [-3 6]

Step 4: Divide each element of the adjoint matrix by the determinant (18).

Inverse = [1/18 2/18] [-3/18 6/18]

Question:

Find the inverse of the matrices , if it exists. $\left[\begin{array}{ccc}1& 3& -2\\ -3& 0& -5\\ 2& 5& 0\end{array}\right]$

Answer:

Step 1: Calculate the determinant of the given matrix.

Determinant = 1*(00) - 3(-52) + (-25*-3) = -25

Step 2: Since the determinant is not 0, the inverse of the matrix exists.

Step 3: Calculate the cofactor matrix.

Cofactor matrix = $\left[\begin{array}{ccc}0& -5& 2\\ 5& -2& 3\\ -3& 0& 1\end{array}\right]$

Step 4: Calculate the adjoint matrix by taking the transpose of the cofactor matrix.

Adjoint matrix = $\left[\begin{array}{ccc}0& 5& -3\\ -5& -2& 0\\ 2& 3& 1\end{array}\right]$

Step 5: Divide the adjoint matrix by the determinant of the given matrix.

Inverse matrix = $\left[\begin{array}{ccc}0& \mathrm{-5/25}& \mathrm{2/25}\\ \mathrm{5/25}& \mathrm{-2/25}& \mathrm{3/25}\\ \mathrm{-3/25}& 0& \mathrm{1/25}\end{array}\right]$

Question:

Find the inverse of the matrices , if it exists. $\left[\begin{array}{cc}2& 1\\ 4& 2\end{array}\right]$

Answer:

Step 1: Rewrite the matrix in the form of $\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]$

$\left[\begin{array}{cc}2& 1\\ 4& 2\end{array}\right]$

Step 2: Calculate the determinant of the matrix.

$\left|\begin{array}{cc}2& 1\\ 4& 2\end{array}\right|=2-8=-6$

Step 3: Check if the determinant is non-zero.

The determinant is -6, which is not zero. Therefore, the inverse of the matrix exists.

Step 4: Calculate the inverse of the matrix using the formula $\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]=*\frac{\left[\begin{array}{cc}\mathrm{d}& -\mathrm{b}\\ -\mathrm{c}& \mathrm{a}\end{array}\right]}{\mathrm{ad}-\mathrm{bc}}$

$\left[\begin{array}{cc}2& 1\\ 4& 2\end{array}\right]=*\frac{\left[\begin{array}{cc}2& -1\\ -4& 2\end{array}\right]}{2-8}$

Step 5: Simplify the inverse matrix.

$[2- ଅଧ୍ୟୟନ ସାମଗ୍ରୀକୁ ଫେରନ୍ତୁ JEE ଅଧ୍ୟୟନ ସାମଗ୍ରୀ (ଗଣିତ) 01 ସମ୍ପର୍କ ଏବଂ କାର୍ଯ୍ୟ ବ୍ୟାୟାମ 01 ବ୍ୟାୟାମ 02 ବ୍ୟାୟାମ 03 02 ଓଲଟା ଟ୍ରାଇଗୋନେଟ୍ରିକ୍ କାର୍ଯ୍ୟଗୁଡ଼ିକ ବ୍ୟାୟାମ 01 ବ୍ୟାୟାମ 02 ବିଭିନ୍ନ ସମାଧାନ ବ୍ୟାୟାମ କରନ୍ତୁ 03 ମ୍ୟାଟ୍ରିକ୍ସ ବ୍ୟାୟାମ 01 ବ୍ୟାୟାମ 02 ବ୍ୟାୟାମ 03 ବ୍ୟାୟାମ 04 ବିବିଧ ସମାଧାନ 04 ନିର୍ଣ୍ଣୟକାରୀ ବ୍ୟାୟାମ 01 ବ୍ୟାୟାମ 02 ବ୍ୟାୟାମ 03 ବ୍ୟାୟାମ 04 ବ୍ୟାୟାମ 05 ବ୍ୟାୟାମ 06 ବିବିଧ ବ୍ୟାୟାମ 05 ନିରନ୍ତରତା ଏବଂ ଭିନ୍ନତା ବ୍ୟାୟାମ 01 ବ୍ୟାୟାମ 02 ବ୍ୟାୟାମ 03 ବ୍ୟାୟାମ 04 ବ୍ୟାୟାମ 05 ବ୍ୟାୟାମ 06 ବ୍ୟାୟାମ 07 ବ୍ୟାୟାମ 08 ବିବିଧ ବ୍ୟାୟାମ 06 ଡେରିଭେଟିକ୍ସର ପ୍ରୟୋଗ ବ୍ୟାୟାମ 01 ବ୍ୟାୟାମ 02 ବ୍ୟାୟାମ 03 ବ୍ୟାୟାମ 04 07 ଇଣ୍ଟିଗ୍ରାଲ୍ ବ୍ୟାୟାମ ବିବିଧ ବ୍ୟାୟାମ 08 ଇଣ୍ଟିଗ୍ରାଲ୍ସର ପ୍ରୟୋଗ ବ୍ୟାୟାମ ବିବିଧ ବ୍ୟାୟାମ 09 ଭେକ୍ଟର୍ ବ୍ୟାୟାମ 01 ବିବିଧ ସମାଧାନ 10 ତିନୋଟି ଡାଇମେନ୍ସନାଲ୍ ଜ୍ୟାମିତି ବ୍ୟାୟାମ 01 ବ୍ୟାୟାମ 02 ବ୍ୟାୟାମ 03 ବିବିଧ ସେନା 11 ରେଖା ପ୍ରୋଗ୍ରାମିଂ ବ୍ୟାୟାମ 01 ବ୍ୟାୟାମ 02 ରେଖା ପ୍ରୋଗ୍ରାମିଂ ବିଭିନ୍ନ ବ୍ୟାୟାମ 12 ସମ୍ଭାବନା ବ୍ୟାୟାମ 01 ବ୍ୟାୟାମ 02 ବ୍ୟାୟାମ 03 ବ୍ୟାୟାମ 04 ବ୍ୟାୟାମ 05 ସମ୍ଭାବ୍ୟ ବିବିଧ ବ୍ୟାୟାମ ଆମ ପାଖରେ ପହଞ୍ଚ | Oriya English Hindi Telugu Punjabi $