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 the product \(AB\) defined?

When the number of columns of \(A\) equals the number of rows of \(B\); if \(A\) is \(m\times n\) and \(B\) is \(n\times p\), then \(AB\) is \(m\times p\).

Is matrix multiplication commutative?

No; in general \(AB\neq BA\). It is, however, associative and distributive over addition.

Define a symmetric matrix.

A square matrix \(A\) is symmetric if \(A^T=A\), i.e. \(a_{ij}=a_{ji}\) for all \(i,j\).

Define a skew-symmetric matrix.

A square matrix \(A\) with \(A^T=-A\); then \(a_{ij}=-a_{ji}\), forcing all diagonal entries to be zero.

State the transpose of a product.

\((AB)^T=B^TA^T\); the transpose of a product reverses the order of the factors.

How is any square matrix split into symmetric and skew parts?

Write \(A=\tfrac{1}{2}(A+A^T)+\tfrac{1}{2}(A-A^T)\); the first part is symmetric and the second skew-symmetric.

What is a scalar matrix?

A diagonal matrix whose diagonal entries are all equal; multiplying it by any conformable matrix scales that matrix.

Give the order of \(AB\) if \(A\) is \(2\times3\) and \(B\) is \(3\times4\).

The product \(AB\) has order \(2\times4\).

What are elementary row operations used for?

They transform a matrix and are used to reduce it to identity form, thereby computing the inverse \(A^{-1}\) when it exists.

What is the identity matrix?

A square matrix \(I\) with \(1\)s on the main diagonal and \(0\)s elsewhere; it satisfies \(AI=IA=A\).

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Chapter 3: Matrices — Class 12 Mathematics

Mathematics · 13 chapters

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Chapter 3: Matrices

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If \(A\) is of order \(2\times3\) and \(B\) is of order \(3\times4\), then \(AB\) has order:

Summary

A matrix is a rectangular array of numbers arranged in rows and columns; its order is \(m\times n\). This chapter develops matrix algebra: equality, addition, scalar multiplication and the row-by-column rule for matrix multiplication, where \((AB)_{ij}=\sum_k a_{ik}b_{kj}\). Matrix multiplication is associative and distributive but generally not commutative, so \(AB\neq BA\) in general. Special matrices are studied — diagonal, scalar, identity, symmetric (\(A^T=A\)) and skew-symmetric (\(A^T=-A\)). The transpose operation and its properties, such as \((AB)^T=B^TA^T\), are established. Elementary row and column operations are introduced and used to find the inverse of a matrix. Matrices provide a compact framework for representing and solving systems of linear equations.

Types and order of matricesMatrix operations and multiplicationTranspose, symmetric and skew-symmetric matricesElementary operations and inverse of a matrix

Key terms

Order of a matrix

The pair \(m\times n\) giving its number of rows and columns.

Matrix multiplication

\((AB)_{ij}=\sum_k a_{ik}b_{kj}\); defined when columns of \(A\) equal rows of \(B\).

Transpose

\(A^T\), obtained by interchanging the rows and columns of \(A\).

Symmetric matrix

A square matrix with \(A^T=A\).

Skew-symmetric matrix

A square matrix with \(A^T=-A\), so its diagonal entries are zero.

Identity matrix

A square matrix \(I\) with ones on the diagonal and zeros elsewhere; \(AI=IA=A\).

Important questions

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Order of a matrix

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The pair \(m\times n\) giving its number of rows and columns.

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