Compute the determinant of the given matrix. If the...

Compute the determinant of the given matrix. If the determinant is nonzero, use the formula for inverting $a 2 \times 2$ matrix to calculate the inverse of the given matrix.
$$
\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]
$$

Key Concepts

2×2 Matrix Inversion Formula

For a 2×2 matrix, there is a specific and convenient formula to compute the inverse. If the matrix is given by [[a, b], [c, d]] and its determinant (ad - bc) is nonzero, the inverse is calculated as (1/(ad - bc)) times the matrix [[d, -b], [-c, a]]. This formula is a direct application of the properties of determinants and matrix algebra specific to the 2×2 case.

Invertibility Condition

A matrix is invertible (or non-singular) if and only if its determinant is nonzero. This condition guarantees the existence of a unique inverse matrix, which when multiplied by the original matrix, results in the identity matrix. Recognizing and verifying invertibility is a fundamental step in solving linear algebra problems involving matrix inverses.

Determinant

The determinant is a scalar value computed from the elements of a square matrix that provides key information about the matrix, such as whether the matrix is invertible. In the context of a 2×2 matrix, the determinant is calculated as ad - bc, using the elements of the matrix arranged as [[a, b], [c, d]]. A nonzero determinant indicates that the matrix has an inverse and that its rows (or columns) are linearly independent.

Matrix Inversion

The inverse of a matrix is the matrix that, when multiplied by the original matrix, yields the identity matrix. For any invertible matrix, finding the inverse is essential for solving systems of linear equations and for many other applications in linear algebra. Inversion provides a systematic way to 'reverse' the effect of the linear transformation defined by the matrix.

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