What traumatic event does Zitkala-Sa describe at the school?

The forcible cutting of her long hair

Why was the cutting of her hair so humiliating to Zitkala-Sa?

Among her people, shingled hair was worn by cowards and mourners

When is a square matrix invertible?

A square matrix \(A\) is invertible if and only if \(|A|\neq0\); such a matrix is called non-singular.

Write the formula for \(A^{-1}\) using the adjoint.

\(A^{-1}=\dfrac{1}{|A|}\,\text{adj }A\), valid only when \(|A|\neq0\).

The cofactor of element \(a_{ij}\) is \(C_{ij}=(-1)^{i+j}M_{ij}\), where \(M_{ij}\) is the corresponding minor.

State the relation \(A(\text{adj }A)\).

\(A(\text{adj }A)=(\text{adj }A)A=|A|\,I\).

How is the area of a triangle found using determinants?

For vertices \((x_1,y_1),(x_2,y_2),(x_3,y_3)\), area \(=\tfrac{1}{2}\left|\,\det\begin{bmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{bmatrix}\right|\).

How are three points shown to be collinear?

They are collinear if the determinant formed by their coordinates (as in the area formula) equals zero.

State the effect of swapping two rows of a determinant.

Interchanging two rows (or columns) changes the sign of the determinant.

Express the solution of \(AX=B\) when \(A\) is invertible.

If \(|A|\neq0\), then \(X=A^{-1}B\), giving a unique solution to the system.

What does \(|AB|\) equal?

\(|AB|=|A|\,|B|\); the determinant of a product equals the product of the determinants.

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:

Flashcards & interactive practice