A scalar that indicates invertibility and scales volume under a transformation.
The determinant is a scalar: ad-bc for 2×2. It is zero exactly when the matrix is singular. Its absolute value is the factor by which areas or volumes scale, and its sign shows orientation changes. Cramer's rule uses determinants to solve small systems.
det [[2,1],[3,4]] = 5.
The determinant tells you whether a system is solvable, and it gives the scaling factor for volumes under a linear transformation.
A scalar that indicates invertibility and scales volume under a transformation.
The determinant tells you whether a system is solvable, and it gives the scaling factor for volumes under a linear transformation.
The matrix is singular. Its columns are linearly dependent, the transformation squashes space, and the system either has no solution or infinitely many.
Use cofactor expansion along a row or column with many zeros, or use row reduction, which is faster for larger matrices.
Typical applications include checking that a stiffness matrix is non-singular before solving, finding the Jacobian determinant for a change of variables in an integral and testing whether a set of control equations is independent.
Study "Matrix operations" first (topic 49 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
det(AB)=det A det B. Swapping rows flips the sign. Cofactor expansion for larger matrices.
det [[2,1],[3,4]] = 5.
The next topic is "Gaussian elimination", which builds directly on this one.
It is classed as Intermediate in Engineering Mathematics (topic 50 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.
Want to practise? Open the interactive course for quick checks, quizzes, simulations and progress tracking.