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.

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