Inverse of a 2 x 2 Matrix
Find the inverse of a 2x2 matrix using (1/det)*[[d,-b],[-c,a]]: compute the determinant, form the adjugate, and multiply by the reciprocal to enable matrix equation solving.
Key Concepts
If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and its determinant $ad cb \neq 0$, then the inverse of $A$ is given by the formula: $$A^{ 1} = \frac{1}{ad cb} \begin{bmatrix} d & b \\ c & a \end{bmatrix}$$.
For $T = \begin{bmatrix} 5 & 2 \\ 4 & 2 \end{bmatrix}$, the determinant is $(5)( 2) (2)( 4) = 2$. So, $T^{ 1} = \frac{1}{ 2} \begin{bmatrix} 2 & 2 \\ 4 & 5 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 2 & 2.5 \end{bmatrix}$. For $M = \begin{bmatrix} 1 & 8 \\ 1 & 9 \end{bmatrix}$, the determinant is $(1)(9) (8)(1) = 1$. So, $M^{ 1} = \frac{1}{1} \begin{bmatrix} 9 & 8 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 9 & 8 \\ 1 & 1 \end{bmatrix}$.
Finding the inverse of a 2x2 matrix is like a magic trick with four simple steps! First, calculate the determinant—it can't be zero. Then, swap the top left and bottom right numbers. After that, change the signs of the other two numbers. Finally, divide the entire new matrix by the determinant you found earlier. Voila, you have the inverse!
Common Questions
What is the formula for the inverse of a 2x2 matrix?
For A=[[a,b],[c,d]], the inverse A-inverse = (1/(ad-bc))*[[d,-b],[-c,a]]. Swap the main diagonal entries, negate the off-diagonal entries, then multiply every entry by 1 divided by the determinant ad-bc. This only works when ad-bc is not zero.
How do you verify that you have correctly computed a matrix inverse?
Multiply A by A-inverse. If the result is the 2x2 identity matrix [[1,0],[0,1]], the inverse is correct. If the product is anything other than the identity matrix, recheck your determinant calculation and the adjugate matrix formation.
How is the inverse matrix used to solve a matrix equation?
For the equation AX=B, multiply both sides on the left by A-inverse: A-inverse*AX = A-inverse*B, which gives IX=A-inverse*B, so X=A-inverse*B. This gives the solution without using elimination or substitution, and is equivalent to Cramer's rule.