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.
$$
\left[\begin{array}{ll}
1 & 0 \\
\frac{3}{4} & 0
\end{array}\right]
$$

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Key Concepts

2x2 Matrix Inversion Formula

For a 2×2 matrix, if the determinant is nonzero, the inverse can be calculated using a specific formula. The formula for the inverse of a matrix [[a, b], [c, d]] is (1/determinant) multiplied by the matrix [[d, -b], [-c, a]], which swaps the diagonal elements and changes the signs of the off-diagonal elements.

Matrix Inversion

Matrix inversion is the process of finding another matrix that, when multiplied with the original matrix, results in the identity matrix. Only non-singular (invertible) matrices, those with nonzero determinants, have inverses.

Singular Matrix

A matrix is said to be singular if its determinant is zero. Singular matrices do not have an inverse because they do not have full rank, meaning that they map the input space into a lower-dimensional space.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether the matrix is invertible and how the matrix scales space. For a 2×2 matrix, the determinant is computed as ad - bc, where the matrix is written in the form [[a, b], [c, d]].

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