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Answers
If a 2×2 matrix A is invertible and is multiplied by its inverse (denoted by the symbol A−1), the resulting product is the Identity matrix which is denoted by II. To illustrate this concept, see the diagram below.
Matrix A when multiplied by its inverse written as A^-1 is equal to the identity matrix I. We can express this is short form as A * A^-1 = I.
In fact, I can switch the order or direction of multiplication between matrices A and A−1, and I would still get the Identity matrix II. That means invertible matrices are commutative.
If A is an invertible matrix then A and its inverse, A^-1, are commutative under the operation of matrix multiplication. That means A times A^-1 and A^-1 times A will have the same product which is the identity matrix I. In the equation forms, we can express these algebraic relationships as A*A^-1=I and A^-1*A=I.
How do we find the inverse of a matrix? The formula is rather simple. As long as you follow it, there shouldn’t be any problem. Here we go.
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The Formula to Find the Inverse of a 2×2 Matrix
Given the matrix A
Matrix A is a 2 by 2 matrix with entries a and b on its first row, and entries c and d on its second row. We can write this in matrix form as A = [a,b;c,d].
Its inverse is calculated using the formula
The formula to find the inverse of an invertible matrix A involves by first calculating the determinant of matrix A and rearranging matrix A from [a,b;c,d] to [d,-b;-c,a]. Now use the scalar value of 1 over determinant of matrix to multiply the rearranged elements of matrix A which is [d,-b;-c,a]. In compact form, the inverse of matrix A or A^-1 = (1/det A) [d,-b;-c,a].
where \color{red}{\rm{det }}\,AdetA is read as the determinant of matrix A.
A few observations about the formula:
Entries \color{blue}aa and \color{blue}dd from matrix A are swapped or interchanged in terms of position in the formula.
Entries \color{blue}bb and \color{blue}cc from matrix A remain in their current positions, however, the signs are reversed. In other words, put negative symbols in front of entries bb and cc.
Since \color{red}{\rm{det }}\,AdetA is just a number, then \large{1 \over {{\rm{det }}A}}
detA
1
is also a number that would serve as the scalar multiplier to the matrix
{[d,-b],[-c,a]}
See my separate lesson on scalar multiplication of matrices.
Examples of How to Find the Inverse of a 2×2 Matrix
Example 1: Find the inverse of the 2×2 matrix below, if it exists.
Matrix A is a 2x2 matrix with entries 5 and 2 on the first row, and entries -7 and -3 on the second row. Therefore, A = [5,2;-7,-3].
The formula requires us to find the determinant of the given matrix. Do you remember how to do that? If not, that’s okay. Review the formula below how to solve for the determinant of a 2×2 matrix.
Let matrix A be A = [a,b;c,d] then its determinant is calculated as det A = (a)(d) minus (b)(c)
So then, the determinant of matrix A is
here's the step by step solution to find the determinant of a 2 by 2 square matrix A with elements 5 and 2 on its first row, and elements -7 a.