In Exercises 13-18, use the fact that if A=[ a b c d ], then A^-1=(1)/(a d-b c)[

step 1 Okay, so in this problem, we need to find the inverse of this matrix A, if possible, which is 6,

In Exercises 13-18, use the fact that if A=[ a b c...

In Exercises $13-18,$ use the fact that if $A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right],$ then $A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}{d} & {-b} \\ {-c} & {a}\end{array}\right]$ to find the inverse of each matrix, if possible. Check that $A A^{-1}=I_{2}$ and $A^{-1} A=I_{2}$ $$ A=\left[\begin{array}{rr}{6} & {-3} \\ {-2} & {1}\end{array}\right] $$

In Exercises $13-18,$ use the fact that if $A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right],$ then $A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}{d} & {-b} \\ {-c} & {a}\end{array}\right]$ to find the inverse of each matrix, if possible. Check that $A A^{-1}=I_{2}$ and $A^{-1} A=I_{2}$
$$
A=\left[\begin{array}{rr}{6} & {-3} \\ {-2} & {1}\end{array}\right]
$$

Key Concepts

Verification of Matrix Inverse

Once the inverse is computed, it can be verified by multiplying the original matrix with its inverse, ensuring that the product in both orders yields the identity matrix. This step confirms the correctness of the computed inverse.

2x2 Matrix Inverse Formula

For a 2x2 matrix A = [[a, b], [c, d]], the inverse is given by A?¹ = (1/(ad - bc))[[d, -b], [-c, a]]. This formula provides a direct method for finding the inverse when the determinant is nonzero.

Identity Matrix

The identity matrix is a special square matrix with ones on the diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix multiplication, meaning that any matrix multiplied by the identity matrix yields the original matrix.

Determinant

The determinant is a scalar value computed from the elements of a square matrix that indicates whether the matrix is invertible. For a 2x2 matrix, the determinant is calculated as ad - bc. The matrix is invertible if and only if the determinant is nonzero.

Matrix Inverse

The matrix inverse of a square matrix A is another matrix, denoted A?¹, such that when multiplied by A, results in the identity matrix. This concept is fundamental in solving matrix equations and is analogous to the reciprocal of a number in real arithmetic.

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