Determine The Invertibility Of Each Of The Following Matrices And Find The Inverse If It Exists.a. A = ( 1 3 0 2 ) A=\begin{pmatrix}1 & 3 \\ 0 & 2\end{pmatrix} A = ( 1 0 ​ 3 2 ​ ) B. B = ( 1 0 0 2 1 0 3 − 2 − 1 ) B=\begin{pmatrix}1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & -2 & -1\end{pmatrix} B = ​ 1 2 3 ​ 0 1 − 2 ​ 0 0 − 1 ​ ​ C.

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Introduction

In linear algebra, matrices play a crucial role in solving systems of equations and representing linear transformations. One of the fundamental concepts in matrix theory is the invertibility of matrices. A matrix is said to be invertible if it has an inverse, which is a matrix that, when multiplied by the original matrix, results in the identity matrix. In this article, we will determine the invertibility of three given matrices and find their inverses if they exist.

Methodology

To determine the invertibility of a matrix, we need to check if its determinant is non-zero. If the determinant is non-zero, the matrix is invertible. We can use the following steps to find the inverse of a matrix:

  1. Check if the matrix is square (i.e., has the same number of rows and columns).
  2. Calculate the determinant of the matrix.
  3. If the determinant is non-zero, the matrix is invertible.
  4. Find the cofactor matrix of the matrix.
  5. Transpose the cofactor matrix to get the adjugate matrix.
  6. Divide the adjugate matrix by the determinant to get the inverse matrix.

Matrix A

Let's start with matrix A:

A=(1302)A=\begin{pmatrix}1 & 3 \\ 0 & 2\end{pmatrix}

Determinant of Matrix A

To determine the invertibility of matrix A, we need to calculate its determinant:

det⁡(A)=(1)(2)−(3)(0)=2\det(A) = (1)(2) - (3)(0) = 2

Since the determinant is non-zero, matrix A is invertible.

Inverse of Matrix A

To find the inverse of matrix A, we need to find the cofactor matrix and then transpose it to get the adjugate matrix. The cofactor matrix of A is:

CA=(20−31)C_A = \begin{pmatrix}2 & 0 \\ -3 & 1\end{pmatrix}

The adjugate matrix is the transpose of the cofactor matrix:

adj(A)=CAT=(2−301)adj(A) = C_A^T = \begin{pmatrix}2 & -3 \\ 0 & 1\end{pmatrix}

Finally, we divide the adjugate matrix by the determinant to get the inverse matrix:

A−1=1det⁡(A)adj(A)=12(2−301)=(1−32012)A^{-1} = \frac{1}{\det(A)}adj(A) = \frac{1}{2}\begin{pmatrix}2 & -3 \\ 0 & 1\end{pmatrix} = \begin{pmatrix}1 & -\frac{3}{2} \\ 0 & \frac{1}{2}\end{pmatrix}

Matrix B

Now, let's move on to matrix B:

B=(1002103−2−1)B=\begin{pmatrix}1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & -2 & -1\end{pmatrix}

Determinant of Matrix B

To determine the invertibility of matrix B, we need to calculate its determinant:

det⁡(B)=(1)∣10−2−1∣−(0)∣203−1∣+(0)∣213−2∣\det(B) = (1)\begin{vmatrix}1 & 0 \\ -2 & -1\end{vmatrix} - (0)\begin{vmatrix}2 & 0 \\ 3 & -1\end{vmatrix} + (0)\begin{vmatrix}2 & 1 \\ 3 & -2\end{vmatrix}

det⁡(B)=(1)(−1)−(0)(2)+(0)(−4)=−1\det(B) = (1)(-1) - (0)(2) + (0)(-4) = -1

Since the determinant is non-zero, matrix B is invertible.

Inverse of Matrix B

To find the inverse of matrix B, we need to find the cofactor matrix and then transpose it to get the adjugate matrix. The cofactor matrix of B is:

CB=(−1002−10−621)C_B = \begin{pmatrix}-1 & 0 & 0 \\ 2 & -1 & 0 \\ -6 & 2 & 1\end{pmatrix}

The adjugate matrix is the transpose of the cofactor matrix:

adj(B)=CBT=(−12−60−12001)adj(B) = C_B^T = \begin{pmatrix}-1 & 2 & -6 \\ 0 & -1 & 2 \\ 0 & 0 & 1\end{pmatrix}

Finally, we divide the adjugate matrix by the determinant to get the inverse matrix:

B−1=1det⁡(B)adj(B)=1−1(−12−60−12001)=(1−2601−200−1)B^{-1} = \frac{1}{\det(B)}adj(B) = \frac{1}{-1}\begin{pmatrix}-1 & 2 & -6 \\ 0 & -1 & 2 \\ 0 & 0 & 1\end{pmatrix} = \begin{pmatrix}1 & -2 & 6 \\ 0 & 1 & -2 \\ 0 & 0 & -1\end{pmatrix}

Conclusion

In this article, we determined the invertibility of three given matrices and found their inverses if they exist. We used the method of calculating the determinant and finding the cofactor matrix to determine the invertibility of the matrices. We also found the inverse of each matrix by dividing the adjugate matrix by the determinant. The results show that matrix A is invertible with an inverse of (1−32012)\begin{pmatrix}1 & -\frac{3}{2} \\ 0 & \frac{1}{2}\end{pmatrix}, and matrix B is invertible with an inverse of (1−2601−200−1)\begin{pmatrix}1 & -2 & 6 \\ 0 & 1 & -2 \\ 0 & 0 & -1\end{pmatrix}.

Introduction

In the previous article, we discussed the concept of matrix invertibility and how to find the inverses of matrices. However, we understand that there may be many questions and doubts that readers may have on this topic. In this article, we will address some of the frequently asked questions (FAQs) on matrix invertibility and inverses.

Q1: What is the difference between a matrix and its inverse?

A1: A matrix is a rectangular array of numbers, while its inverse is a matrix that, when multiplied by the original matrix, results in the identity matrix. The identity matrix is a special matrix that has 1s on the main diagonal and 0s elsewhere.

Q2: How do I know if a matrix is invertible?

A2: To determine if a matrix is invertible, you need to calculate its determinant. If the determinant is non-zero, the matrix is invertible. If the determinant is zero, the matrix is not invertible.

Q3: What is the determinant of a matrix?

A3: The determinant of a matrix is a scalar value that can be calculated using various methods, including the expansion by minors method, the cofactor expansion method, and the Laplace expansion method.

Q4: How do I find the inverse of a matrix?

A4: To find the inverse of a matrix, you need to follow these steps:

  1. Check if the matrix is square (i.e., has the same number of rows and columns).
  2. Calculate the determinant of the matrix.
  3. If the determinant is non-zero, the matrix is invertible.
  4. Find the cofactor matrix of the matrix.
  5. Transpose the cofactor matrix to get the adjugate matrix.
  6. Divide the adjugate matrix by the determinant to get the inverse matrix.

Q5: What is the cofactor matrix of a matrix?

A5: The cofactor matrix of a matrix is a matrix that contains the cofactors of the original matrix. The cofactors are calculated by finding the determinant of the submatrix formed by removing the row and column of the element and then multiplying it by a sign.

Q6: What is the adjugate matrix of a matrix?

A6: The adjugate matrix of a matrix is the transpose of the cofactor matrix. It is also known as the classical adjugate.

Q7: How do I calculate the determinant of a 3x3 matrix?

A7: To calculate the determinant of a 3x3 matrix, you can use the expansion by minors method. This involves finding the determinant of the 2x2 submatrices formed by removing the row and column of the element and then multiplying it by a sign.

Q8: What is the significance of the inverse of a matrix?

A8: The inverse of a matrix is significant because it can be used to solve systems of linear equations. If a matrix A represents a system of linear equations, then its inverse A^-1 can be used to find the solution to the system.

Q9: Can a matrix have multiple inverses?

A9: No, a matrix cannot have multiple inverses. The inverse of a matrix is unique and is denoted by A^-1.

Q10: How do I check if a matrix is invertible using a calculator?

A10: To check if a matrix is invertible using a calculator, you can use the following steps:

  1. Enter the matrix into the calculator.
  2. Press the "det" or "determinant" button to calculate the determinant of the matrix.
  3. If the determinant is non-zero, the matrix is invertible.

Conclusion

In this article, we addressed some of the frequently asked questions (FAQs) on matrix invertibility and inverses. We hope that this article has provided a clear understanding of the concepts and has helped to clarify any doubts that readers may have had.