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.

CBSE · Senior secondary

CBSE Class 12 — Notes, Chapters & Practice Quizzes

Master every Class 12 chapter — the base your board marks and your NEET / JEE / CUET prep both stand on.

13 subjects 166 chapters 171 practice quizzes NCERT syllabus

56 quiz attempts so far Dual AI-verified questions 10 quizzes free, then 10 credits

Notice Board

Students preparing for CBSE Class 12 Board Examination

View all

Formula of the day

Physics · EM Induction · 13 formulae in the cheatsheet

Trap of the day

Lens/mirror sign convention: distances against the incident light are negative. Drop a sign and real becomes virtual.

Chapter 4: Determinants — Class 12 Mathematics

Mathematics · 13 chapters

Summary, key terms, important questions and a practice quiz with AI diagnosis for each.

Chapter 4: Determinants

Try one from this chapter

The determinant of \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\) is:

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 \(|A|\neq0\).

Important questions

Explore interactively

Key-term flashcards

Flip cards · mark known · keyboard ← → and Space

6 cards

Term1 / 6

Determinant

Tap to reveal

Meaning1 / 6

A scalar \(|A|\) associated with a square matrix; for \(2\times2\), \(ad-bc\).

Tap to flip back

Tap card to flip

Keyboard: ← → to move · Space to flip

Practice quiz · Determinants

Score on this chapter, climb the leaderboard, and get an AI diagnosis of your mistakes.

Dual AI-verified questions Real exam pattern 2 quizzes free, then 10 credits per quiz

Determinants

Maths 10 Qs · ~10 min

7/10avg score

7/10best

CBSE Class 12 — Matrices and Determinants

10 Qs · ~10 min