Multiplicative inverse of a square matrix
Find the multiplicative inverse of a 2×2 matrix in Grade 10 algebra using the formula A⁻¹ = (1/det(A))·[d,-b;-c,a], verifying that A·A⁻¹ equals the identity matrix.
Key Concepts
The inverse of a matrix $A$, notated $A^{ 1}$, is a special matrix where multiplying it by $A$ results in the identity matrix, $I$. For two square matrices $A$ and $B$ to be inverses of each other, their product must be the identity matrix, regardless of the order of multiplication: $A \cdot B = B \cdot A = I$.
Are $P = \begin{bmatrix} 3 & 7 \\ 2 & 5 \end{bmatrix}$ and $Q = \begin{bmatrix} 5 & 7 \\ 2 & 3 \end{bmatrix}$ inverses? Yes, because $P \cdot Q = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I$. Are $R = \begin{bmatrix} 1 & 1 \\ 2 & 3 \end{bmatrix}$ and $S = \begin{bmatrix} 3 & 1 \\ 2 & 0 \end{bmatrix}$ inverses? No, because $R \cdot S = \begin{bmatrix} 1 & 1 \\ 0 & 2 \end{bmatrix} \neq I$.
Think of it like a secret decoder ring! Multiplying a matrix by its inverse is like undoing a secret code, getting you back to the simple identity matrix, which is the matrix equivalent of the number 1. This 'undo' feature is super powerful for solving complex matrix equations, just like division undoes multiplication with regular numbers.
Common Questions
What is the formula for the inverse of a 2×2 matrix?
For A=[[a,b],[c,d]], A⁻¹ = (1/(ad-bc))·[[d,-b],[-c,a]]. The scalar factor is 1/det(A).
How do you verify a matrix inverse is correct?
Multiply A·A⁻¹. The result should equal the 2×2 identity matrix [[1,0],[0,1]]. If it does, the inverse is correct.
When does a 2×2 matrix have no inverse?
When its determinant ad-bc = 0, the matrix is singular and has no inverse. This corresponds to a system with no unique solution.