Matrices
Class 12 Mathematics · Mathematics Part I · Chapter 3
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Physics
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Economics
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History
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History
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History
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Geography
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Geography
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Geography
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Psychology
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English Core · Flamingo
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Home Science
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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; \(AI=IA=A\).
Extra questions & answers
Try one from this chapter
If \(A\) is of order \(2\times3\) and \(B\) is of order \(3\times4\), then \(AB\) has order: