Question

Please help!!

Give an example of a 2 by 2 matrix and describe how to find the inverse of the matrix. When does a matrix not have an inverse?

Answer 1

A 2 by 2 matrix can be represented as [[a, b], [c, d]]. The inverse of a matrix exists if the determinant (ad - bc) is non-zero. To find the inverse, swap elements a and d, negate b and c, and then divide each element by the determinant.

Step-by-step explanation:

An example of a 2 by 2 matrix is:

\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]

To find the inverse of this matrix, denoted as \( A^{-1} \), you should follow these steps:

1. 
   
3. Calculate the determinant of matrix A, which is \( ad - bc \).
4. 
   
6. If the determinant is not zero, you can proceed. The matrix has an inverse if and only if its determinant is non-zero.
7. 
   
9. Swap the elements \( a \) and \( d \).
10. 
    
12. Change the signs of \( b \) and \( c \).
13. 
    
15. Divide each term of the resulting matrix by the determinant.

The inverse is thus given by:

\[ A^{-1} = \frac{1}{{ad - bc}} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]

A matrix does not have an inverse if its determinant is zero because it would mean there is no scalar you can multiply the matrix by to get the identity matrix.

Answer 2

the last line answer ur second qn and the above ate examples