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- స్టడీ మెటీరియల్‌కి తిరిగి వెళ్ళు JEE స్టడీ మెటీరియల్ (గణితం) 01 సంబంధాలు మరియు విధులు వ్యాయామం 01 వ్యాయామం 02 వ్యాయామం 03 02 విలోమ త్రికోణమితి విధులు వ్యాయామం 01 వ్యాయామం 02 ఇతర పరిష్కారాలను వ్యాయామం చేయండి 03 మాత్రికలు వ్యాయామం 01 వ్యాయామం 02 వ్యాయామం 03 వ్యాయామం 04 ఇతర పరిష్కారాలు 04 నిర్ణాయకాలు వ్యాయామం 01 వ్యాయామం 02 వ్యాయామం 03 వ్యాయామం 04 వ్యాయామం 05 వ్యాయామం 06 ఇతర వ్యాయామాలు 05 కొనసాగింపు మరియు భిన్నత్వం వ్యాయామం 01 వ్యాయామం 02 వ్యాయామం 03 వ్యాయామం 04 వ్యాయామం 05 వ్యాయామం 06 వ్యాయామం 07 వ్యాయామం 08 ఇతర వ్యాయామాలు 06 డెరివేటివ్‌ల అప్లికేషన్ వ్యాయామం 01 వ్యాయామం 02 వ్యాయామం 03 వ్యాయామం 04 07 సమగ్రతలు ఎజక్సర్సైజ్ ఇతరత్రా వ్యాయామం 08 ఇంటిగ్రల్స్ యొక్క అప్లికేషన్ ఎజక్సర్సైజ్ ఇతరత్రా వ్యాయామం 09 వెక్టర్స్ వ్యాయామం 01 ఇతర పరిష్కారాలు 10 త్రీ డైమెన్షనల్ జ్యామితి వ్యాయామం 01 వ్యాయామం 02 వ్యాయామం 03 ఇతర సైన్యం 11 లీనియర్ ప్రోగ్రామింగ్ వ్యాయామం 01 వ్యాయామం 02 లీనియర్ ప్రోగ్రామింగ్ ఇతర వ్యాయామాలు 12 సంభావ్యత వ్యాయామం 01 వ్యాయామం 02 వ్యాయామం 03 వ్యాయామం 04 వ్యాయామం 05 సంభావ్యత ఇతర వ్యాయామం మమ్మల్ని చేరుకోండి $