Consider the 2 by 2 square matrix A=[ a b c d ] If D=a d-b c ≠0, show that A i

step 1 So we're given this problem where we have a matrix A, and we are told that the determinant D is

Consider the 2 by 2 square matrix A=[ a b c ...

Consider the 2 by 2 square matrix
$$
A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]
$$
If $D=a d-b c \neq 0$, show that $A$ is nonsingular and that
$$
A^{-1}=\frac{1}{D}\left[\begin{array}{rr}
d & -b \\
-c & a
\end{array}\right]
$$

Key Concepts

Adjugate Matrix

The adjugate (or classical adjoint) of a matrix is formed by taking the transpose of the cofactor matrix. In the case of a 2×2 matrix, the adjugate is computed by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. This construction plays an integral role in the formula for finding the inverse of a matrix, especially when the determinant is nonzero.

Determinant

The determinant is a scalar value derived from a square matrix that provides important information about the matrix, such as whether it has an inverse. For a 2×2 matrix, the determinant is calculated as ad - bc. A nonzero determinant indicates that the matrix is invertible, while a zero determinant implies that the matrix is singular and does not have an inverse.

Nonsingular Matrix

A nonsingular matrix is a square matrix that has an inverse. The key condition for a matrix to be nonsingular is that its determinant must not equal zero. This characteristic is fundamental in linear algebra as it ensures the matrix represents a bijective linear transformation where every input corresponds to a unique output.

Inverse of a Matrix

The inverse of a matrix A, denoted by A?¹, is a matrix that, when multiplied with A, results in the identity matrix. The computation of this inverse is critical for solving systems of linear equations and for various applications in mathematics and engineering. For a 2×2 matrix, the inverse is given by scaling the adjugate matrix by the reciprocal of the determinant.

Transcript

Please give Ace some feedback