Matrix notation, dimensions, addition, and scalar multiplication

Key idea

A matrix is an array of numbers. Its dimension is \(m\times n\) (rows \(\times\) columns). Two matrices can be added only if they have the same size. Scalar multiplication multiplies every entry by the scalar.

Definitions you’ll use constantly

• Equality: \(A=B\) if they have the same size and every corresponding entry matches.
• Addition: \((A+B)_{ij}=a_{ij}+b_{ij}\) (same size required).
• Scalar multiplication: \((kA)_{ij}=k\,a_{ij}\).

Worked example

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Summary

• Matrix addition requires equal sizes; scalar multiplication always works.
• Do arithmetic entry-by-entry when adding or scaling.

Matrix multiplication, dimension checks, and the identity matrix

Key idea

If \(A\) is \(m\times n\) and \(B\) is \(n\times p\), then \(AB\) is defined and is \(m\times p\). Each entry of \(AB\) is a row-by-column dot product: \[ (AB)_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}. \] In general, AB≠ BA even when both products are defined.

The identity matrix

The identity matrix \(I_n\) has ones on the diagonal and zeros elsewhere. It acts like “1” for multiplication: \[ I_nA=A,\quad AI_n=A. \]

Worked example

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Summary

• Matrix multiplication is row-by-column and requires matching inner dimensions.
• \(I_n\) acts as the multiplicative identity; matrix multiplication is usually not commutative.

Transpose, symmetry, and trace

Transpose rules

The transpose flips a matrix across its diagonal: rows become columns. Key properties:

• \((A^T)^T=A\)
• \((A+B)^T=A^T+B^T\)
• \((AB)^T=B^TA^T\)

Symmetry and trace

• Symmetric matrix: \(A\) is symmetric if \(A=A^T\).
• Trace: for a square matrix, \(\mathrm{tr}(A)\) is the sum of diagonal entries.

Worked example

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Summary

• Transpose swaps rows and columns; symmetry means \(A=A^T\).
• The trace is the sum of diagonal entries of a square matrix.

Determinants: \(2\times 2\) formula, triangular shortcut, and key properties

The \(2\times 2\) determinant

For a \(2\times 2\) matrix, \[ \det\begin{pmatrix}a & b\\c & d\end{pmatrix}=ad-bc. \] A square matrix is invertible exactly when its determinant is not zero.

Fast shortcut: triangular matrices

If a matrix is upper triangular or lower triangular, its determinant is the product of diagonal entries. For example, \[ \det\begin{pmatrix}2 & 3\\0 & 4\end{pmatrix}=2\cdot 4=8. \]

Two must-know properties

• \(\det(A^T)=\det(A)\)
• \(\det(AB)=\det(A)\det(B)\) (when both products make sense and \(A,B\) are square of the same size)

Worked example

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Summary

• For \(2\times 2\), \(\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\).
• Triangular determinant = product of the diagonal.
• Invertible \Leftrightarrow \det(A)≠ 0.

Matrix inverses: the \(2\times 2\) inverse formula and solving \(Ax=b\)

Key idea

A square matrix \(A\) is invertible if there exists \(A^{-1}\) such that \[ A^{-1}A=AA^{-1}=I. \] For a \(2\times 2\) matrix \(A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\), if \det(A)=ad-bc≠ 0, then \[ A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}. \]

Worked example

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Summary

• A matrix is invertible exactly when \det(A)≠ 0.
• For \(2\times 2\), use \(\dfrac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\).
• When \(A^{-1}\) exists, \(Ax=b\) has the solution \(x=A^{-1}b\).

Invertibility checks and clean algebra with transposes and inverses

Key properties to memorize

• Transpose and determinant: \(\det(A^T)=\det(A)\).
• Transpose and inverse: if \(A\) is invertible, \((A^T)^{-1}=(A^{-1})^T\).
• Product inverse: if \(A,B\) are invertible, \((AB)^{-1}=B^{-1}A^{-1}\) (reverse order!).
• Symmetry and inverses: if \(A\) is symmetric and invertible, then \(A^{-1}\) is symmetric.

Worked example: a quick invertibility check

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Summary

• \det(A)≠ 0 is the fastest invertibility check (especially for \(2\times 2\)).
• Remember the “reverse order” rule: \((AB)^{-1}=B^{-1}A^{-1}\).
• Symmetric + invertible \(\Rightarrow\) inverse is symmetric.

Why matrix arithmetic and inverses matter

Where these skills show up

• Solving systems: \(Ax=b\) uses matrix multiplication, determinants, and inverses.
• Linear transformations: matrices represent scaling, rotation, shear, and reflections.
• Geometry and area scaling: determinants describe how area (or volume) scales under a transformation.
• Data and computation: efficient matrix operations power graphics, optimization, and machine learning.

Worked example: diagonal determinants

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Final recap

• Dimensions: check sizes before adding or multiplying.
• Multiplication: row-by-column, and AB≠ BA in general.
• Transpose: swap rows/columns; \((AB)^T=B^TA^T\).
• Trace: sum of diagonal entries.
• Determinant: for \(2\times 2\), \(ad-bc\); triangular shortcut = product of diagonal.
• Invertibility: \(A^{-1}\) exists exactly when \det(A)≠ 0.
• \(2\times 2\) inverse: \(\dfrac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\).