Is a matrix invertible?

Learning Objectives Define an invertible (non-singular) matrix and a singular matrix. Calculate the determinant of a 2x2 square matrix. Calculate the determinant of a 3x3 square matrix using cofactor expansion. State the fundamental condition for a matrix to be invertible. Use the determinant to determine if a given square matrix is invertible or singular. Explain the relationship between a matrix's determinant and its invertibility. Ever tried to 'undo' a command on your computer? 🤔 An inverse matrix does the same for geometric transformations, but not every matrix has an 'undo' button! This tutorial will teach you how to determine if a matrix has an inverse. We will focus on a special number called the determinant, which is the key to unlocking t...

TermDefinitionExample Square MatrixA matrix with an equal number of rows and columns (e.g., 2x2, 3x3). Only square matrices can be invertible.A = [[5, 1], [2, 3]] is a 2x2 square matrix. Invertible Matrix (Non-singular)A square matrix 'A' is invertible if there exists another matrix, 'A⁻¹', such that their product is the identity matrix (A * A⁻¹ = I). An invertible matrix has a non-zero determinant.If A = [[2, 1], [1, 1]], its inverse is A⁻¹ = [[1, -1], [-1, 2]]. Singular MatrixA square matrix that does not have an inverse. A matrix is singular if and only if its determinant is zero.B = [[1, 2], [2, 4]] is singular because you cannot find a matrix B⁻¹ such that B * B⁻¹ = I. Determinant (det(A) or |A|)A unique scalar (a single number) calculated from the elements of a s...

The Invertibility Condition A square matrix A is invertible if and only if det(A) \neq 0. This is the most important rule. If the determinant is any number other than zero, the matrix has an inverse. If the determinant is exactly zero, the matrix is singular and has no inverse. Determinant of a 2x2 Matrix For A = [[a, b], [c, d]], the determinant is det(A) = ad - bc. To find the determinant of a 2x2 matrix, multiply the elements on the main diagonal (a*d) and subtract the product of the elements on the other diagonal (b*c). Determinant of a 3x3 Matrix (Cofactor Expansion) For A = [[a, b, c], [d, e, f], [g, h, i]], det(A) = a(ei - fh) - b(di - fg) + c(dh - eg). This formula breaks the 3x3 determinant down into three 2x2 determinant calculations. Note the minus sign on...