Determinants
Class 12 Mathematics · Mathematics Part I · Chapter 4
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Summary
The determinant is a scalar associated with a square matrix that captures key information about it. For a \(2\times2\) matrix \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\) the determinant is \(ad-bc\); for larger matrices it is computed by expansion along a row or column using minors and cofactors. This chapter covers properties of determinants, the use of determinants to find the area of a triangle, the adjoint (\(\text{adj }A\)) of a matrix, and the inverse via \(A^{-1}=\dfrac{1}{|A|}\,\text{adj }A\), valid when \(|A|\neq0\). A matrix is invertible (non-singular) exactly when its determinant is non-zero. Determinants are then applied to solve systems of linear equations using Cramer's-style matrix methods (\(AX=B\Rightarrow X=A^{-1}B\)), linking algebra to geometry and to consistency of equations.
Determinant of a matrixProperties of determinantsMinors, cofactors and adjointInverse of a matrixSolving linear systems
Key terms
Determinant
A scalar \(|A|\) associated with a square matrix; for \(2\times2\), \(ad-bc\).
Minor
The determinant left after deleting the row and column of an element.
Cofactor
A signed minor: \(C_{ij}=(-1)^{i+j}M_{ij}\).
Adjoint
\(\text{adj }A\), the transpose of the matrix of cofactors.
Singular matrix
A square matrix with \(|A|=0\); it has no inverse.
Inverse via adjoint
\(A^{-1}=\dfrac{1}{|A|}\,\text{adj }A\) when \(|A|\neq0\).
Extra questions & answers
Try one from this chapter
The determinant of \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\) is: