Thursday, February 22, 2024
HomematrixInverse of a matrix solved example 01

Inverse of a matrix solved example 01

Inverse of a matrix solved example 01. In this post, we will solve one example and provide step by step solution.

Inverse of a matrix solved example 01

Given matrix:
A=\begin{bmatrix}2 & 1 & 4 \\3 & 2 & 6 \\3 & 5 & 3 \\\end{bmatrix}

  1. Calculate the determinant of the matrix to find the inverse of a matrix:

    \text{Det} = 2 \cdot (2 \cdot 3 - 6 \cdot 5) - 1 \cdot (3 \cdot 3 - 6 \cdot 3) + 4 \cdot (3 \cdot 5 - 2 \cdot 3) = -52
  2. Calculate the matrix of minors:
    \text{Minors} =\begin{bmatrix}6 & -12 & 3 \\-3 & 6 & -3 \\-4 & 6 & 8 \\\end{bmatrix}
  3. Calculate the matrix of cofactors (alternating signs):
    \text{Cofactors} =\begin{bmatrix}6 & 12 & 3 \\3 & 6 & -3 \\-4 & -6 & 8 \\\end{bmatrix}
  4. Transpose the matrix of cofactors to get the adjugate matrix:
    \text{Adjugate} =\begin{bmatrix}6 & 3 & -4 \\12 & 6 & -6 \\3 & -3 & 8 \\\end{bmatrix}
  5. Calculate the inverse by dividing the adjugate matrix by the determinant:
    \text{Inverse} = \frac{1}{\text{Det}} \cdot \text{Adjugate} =\begin{bmatrix}-0.115 & -0.058 & 0.077 \\-0.231 & -0.115 & 0.115 \\-0.058 & 0.058 & -0.154 \\\end{bmatrix}
Inverse of a matrix solved example 02

Inverse of a matrix solved example 02

Given matrix:

A =\begin{bmatrix}1 & 0 & 2 \\0 & 2 & 1 \\1 & -1 & 1 \\\end{bmatrix}

  1. Calculate the determinant of the matrix:

    \text{Det} = 1 \cdot (2 \cdot 1 - 1 \cdot (-1)) - 0 \cdot (0 \cdot 1 - 1 \cdot 1) + 2 \cdot (0 \cdot (-1) - 2 \cdot 1) = 5
  2. Calculate the matrix of minors:

    \text{Minors} =\begin{bmatrix}2 & 1 & 2 \\-1 & 1 & 1 \\-2 & -2 & 2 \\\end{bmatrix}
  3. Calculate the matrix of cofactors (alternating signs):

    \text{Cofactors} =\begin{bmatrix}2 & -1 & 2 \\1 & 1 & -1 \\-2 & 2 & 2 \\\end{bmatrix}
  4. Transpose the matrix of cofactors to get the adjugate matrix:

    \text{Adjugate} =\begin{bmatrix}2 & 1 & -2 \\-1 & 1 & 2 \\2 & -1 & 2 \\\end{bmatrix}
  5. Calculate the inverse by dividing the adjugate matrix by the determinant:
    \text{Inverse} = \frac{1}{\text{Det}} \cdot \text{Adjugate} =\begin{bmatrix}0.4 & 0.2 & -0.4 \\-0.2 & 0.2 & 0.4 \\0.4 & -0.2 & 0.4 \\\end{bmatrix}

Inverse of a matrix solved example

Given matrix:

A =\begin{bmatrix}5 & 2 & -3 \\3 & -1 & 1 \\-2 & 1 & 2 \\\end{bmatrix}

  1. Calculate the determinant of the matrix:

    \text{Det} = 5 \cdot (-1 \cdot 2 - 1 \cdot 1) - 2 \cdot (3 \cdot 2 - 1 \cdot -2) - 3 \cdot (3 \cdot 1 - (-1) \cdot -2) = -2
  2. Calculate the matrix of minors:

    \text{Minors} =\begin{bmatrix}1 & -7 & 1 \\-2 & -8 & -2 \\-1 & -1 & -5 \\\end{bmatrix}
  3. Calculate the matrix of cofactors (alternating signs):

    \text{Cofactors} =\begin{bmatrix}1 & 7 & 1 \\2 & -8 & 2 \\-1 & 1 & -5 \\\end{bmatrix}
  4. Transpose the matrix of cofactors to get the adjugate matrix:

    \text{Adjugate} =\begin{bmatrix}1 & 2 & -1 \\7 & -8 & 1 \\1 & 2 & -5 \\\end{bmatrix}
  5. Calculate the inverse by dividing the adjugate matrix by the determinant:

    \text{Inverse} = \frac{1}{\text{Det}} \cdot \text{Adjugate} =\begin{bmatrix}-\frac{1}{2} & -1 & \frac{1}{2} \\-\frac{7}{2} & 4 & -\frac{1}{2} \\-\frac{1}{2} & -1 & \frac{5}{2} \\\end{bmatrix}
Inverse of a matrix solved example 03
Inverse of a matrix solved example 03

Inverse of a matrix solved example 03

Given matrix:

A =\begin{bmatrix}6 & 7 & 8 \\3 & 4 & 5 \\2 & 3 & 4 \\\end{bmatrix}

  1. Calculate the determinant of the matrix:

    \text{Det} = 6 \cdot (4 \cdot 4 - 5 \cdot 3) - 7 \cdot (3 \cdot 4 - 5 \cdot 2) + 8 \cdot (3 \cdot 3 - 4 \cdot 2) = 1
  2. Calculate the matrix of minors:

    \text{Minors} =\begin{bmatrix}7 & 6 & 3 \\5 & 4 & 2 \\3 & 2 & 1 \\\end{bmatrix}
  3. Calculate the matrix of cofactors (alternating signs):

    \text{Cofactors} =\begin{bmatrix}7 & -6 & 3 \\-5 & 4 & -2 \\3 & -2 & 1 \\\end{bmatrix}
  4. Transpose the matrix of cofactors to get the adjugate matrix:

    \text{Adjugate} =\begin{bmatrix}7 & -5 & 3 \\-6 & 4 & -2 \\3 & -2 & 1 \\\end{bmatrix}
  5. Calculate the inverse by dividing the adjugate matrix by the determinant:

    \text{Inverse} = \frac{1}{\text{Det}} \cdot \text{Adjugate} =\begin{bmatrix}7 & -5 & 3 \\-6 & 4 & -2 \\3 & -2 & 1 \\\end{bmatrix}
Inverse of a matrix solved example 04
Inverse of a matrix solved example 04

Inverse of a matrix solved example 04

Given matrix:

A=\begin{bmatrix}1 & 0 & 2 \\0 & 2 & 1 \\1 & -1 & 1 \\\end{bmatrix}

  1. Calculate the determinant of the matrix:

\text{Det} = 1 \cdot (2 \cdot 1 - 1 \cdot (-1)) - 0 \cdot (0 \cdot 1 - 1 \cdot 1) + 2 \cdot (0 \cdot (-1) - 2 \cdot 1) = 5

  1. Calculate the matrix of minors:

\text{Minors} =\begin{bmatrix}2 & 1 & 2 \\-1 & 1 & 1 \\-2 & -2 & 2 \\\end{bmatrix}

  1. Calculate the matrix of cofactors (alternating signs):

\text{Cofactors} =\begin{bmatrix}2 & -1 & 2 \\1 & 1 & -1 \\-2 & 2 & 2 \\\end{bmatrix}

  1. Transpose the matrix of cofactors to get the adjugate matrix:

\text{Adjugate} =\begin{bmatrix}2 & 1 & -2 \\-1 & 1 & 2 \\2 & -1 & 2 \\\end{bmatrix}

  1. Calculate the inverse by dividing the adjugate matrix by the determinant:

*** QuickLaTeX cannot compile formula:
\text{Inverse} = \frac{1}{\text{Det}} \cdot \text{Adjugate} =\begin{bmatrix}0.4 & 0.2 & -0.4 \\
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<figure class="wp-block-image size-large"><img src="https://apkplot.com/wp-content/uploads/2023/08/Inverse-of-the-Matrices-Ex-05-1024x536.png" alt="Inverse of a matrix solved example 05" class="wp-image-840"/><figcaption class="wp-element-caption">Inverse of a matrix solved example 05</figcaption></figure>
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<h2 class="wp-block-heading">Inverse of a matrix solved example 05</h2>
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<!-- wp:paragraph -->
Given matrix:

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A =\begin{bmatrix}
5 & 2 & -3 \\
3 & -1 & 1 \\
-2 & 1 & 2 \\
\end{bmatrix} <!-- /wp:paragraph -->  <!-- wp:list {"ordered":true} --> <ol><!-- wp:list-item --> <li>Calculate the <a href="https://apkplot.com/inverse-of-a-matrix-solved-example-04/" data-type="post" data-id="559">determinant of the matrix</a>:\text{Det} = 5 \cdot (-1 \cdot 2 – 1 \cdot 1) – 2 \cdot (3 \cdot 2 – 1 \cdot -2) – 3 \cdot (3 \cdot 1 – (-1) \cdot -2) = -2</li> <!-- /wp:list-item -->  <!-- wp:list-item --> <li>Calculate the matrix of minors:\text{Minors} =
\begin{bmatrix}
1 & -7 & 1 \\
-2 & -8 & -2 \\
-1 & -1 & -5 \\
\end{bmatrix}</li> <!-- /wp:list-item -->  <!-- wp:list-item --> <li>Calculate the matrix of cofactors (alternating signs):\text{Cofactors} =
\begin{bmatrix}
1 & 7 & 1 \\
2 & -8 & 2 \\
-1 & 1 & -5 \\
\end{bmatrix}</li> <!-- /wp:list-item -->  <!-- wp:list-item --> <li><a href="http://www.pakmath.com" data-type="link" data-id="www.pakmath.com">Transpose the matrix</a> of cofactors to get the adjugate matrix:\text{Adjugate} =
\begin{bmatrix}
1 & 2 & -1 \\
7 & -8 & 1 \\
1 & 2 & -5 \\
\end{bmatrix}</li> <!-- /wp:list-item -->  <!-- wp:list-item --> <li>Calculate the inverse by dividing the adjugate matrix by the determinant:\text{Inverse} = \frac{1}{\text{Det}} \cdot \text{Adjugate} =
\begin{bmatrix}
-\frac{1}{2} & -1 & \frac{1}{2} \\
-\frac{7}{2} & 4 & -\frac{1}{2} \\
-\frac{1}{2} & -1 & \frac{5}{2} \\
\end{bmatrix}$
apkplot
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Apkplot.com is an all-in-one educational site that has answers to math problems and free past papers and MCQs to help people prepare for the MDCAT and ECAT. The website is based on FSC and covers the subjects of math, physics, chemistry, and biology for Parts 01 and 02. Students can use the website's large database of answers to math problems and step-by-step explanations to help them. Students taking FSC Parts 01 and 02, as well as the MCAT and ECAT, can also find past papers on the website. These papers help students understand the format of the exam and prepare well for the real test. The MCQs are also a great way for students to test their knowledge and figure out what they need to learn more about.
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