Matrices

The dimensions, order or size of a matrix are written rows by columns. An $m\times n$ matrix has $m$ rows and $n$ columns.

Two matrices are equal only when they have the same dimensions and every pair of corresponding elements is equal.

Equality of matrices can create several scalar equations. Equate corresponding positions, then solve those real equations.

Dimensions are the first compatibility check before any matrix operation. Write them beside the matrices before calculating.

Matrices can be added or subtracted only when they have the same dimensions. Combine corresponding entries in the same positions.

Addition is commutative and associative: $A+B=B+A$ and $(A+B)+C=A+(B+C)$, provided all matrices have compatible dimensions.

Subtraction is not commutative. Distribute the negative sign to every element of the second matrix.

In models, addition can combine data tables with identical row and column meanings; subtraction can measure change between two time periods.

A scalar is an ordinary number. Multiplying a matrix by a scalar multiplies every element by that number without changing the dimensions.

The zero matrix $O_{m\times n}$ has zero in every position. It is the additive identity because $A+O=A$ for matching dimensions.

Scalar multiplication distributes over matrix addition: $k(A+B)=kA+kB$.

In applications, scalar multiplication changes every entry by the same scale factor, such as converting costs from dollars to thousands of dollars.

The product $AB$ exists 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 $m\times p$.

Each entry of the product is a row-column dot product. Multiply corresponding entries of a row of $A$ and a column of $B$, then add the results.

Matrix multiplication is generally not commutative: $AB$ may differ from $BA$, and one order may exist when the other does not.

Multiplication represents composition or aggregation. In a model, the units and meanings of the inner dimensions must match, not just their numerical sizes.

The identity matrix $I_n$ is the square matrix with ones on its main diagonal and zeros elsewhere. It leaves compatible matrices unchanged: $AI=IA=A$.

For a square matrix, $A^2=AA$ and $A^n$ means repeated matrix multiplication. It does not mean raising each element separately.

Matrix powers describe repeated transformations or repeated state transitions. Technology is especially useful for large exponents.

The zero power of an invertible square matrix is $A^0=I$, matching ordinary exponent conventions.

The determinant is a scalar associated with a square matrix. For $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, the determinant is $ad-bc$.

A square matrix is singular when its determinant is zero. A singular matrix has no inverse and collapses area or volume in its geometric interpretation.

For a two-dimensional transformation, $|\det A|$ is the area scale factor. A negative determinant also reverses orientation.

For larger matrices, use the approved GDC when permitted and report enough working to identify the matrix and operation.

The inverse $A^{-1}$ reverses the effect of a non-singular square matrix: $AA^{-1}=A^{-1}A=I$.

For a $2\times2$ matrix, swap the diagonal entries, change the signs of the off-diagonal entries, and divide by the determinant.

The inverse exists only when the determinant is non-zero. Never divide by a zero determinant.

Order remains important with inverses: $(AB)^{-1}=B^{-1}A^{-1}$. The order reverses because the last transformation must be undone first.

Matrix equations resemble algebraic equations, but multiplication order must be preserved. From $AX=B$, premultiply by $A^{-1}$ to obtain $X=A^{-1}B$.

From $XA=B$, postmultiply by $A^{-1}$ to obtain $X=BA^{-1}$. These answers are generally different.

An equation may also be solved elementwise by assigning unknown entries to $X$ and equating corresponding elements.

Always state the invertibility condition for any matrix whose inverse is used.

A linear system can be written compactly as $A\mathbf{x}=\mathbf{b}$, where $A$ is the coefficient matrix, $\mathbf{x}$ is the unknown column vector and $\mathbf{b}$ is the constants vector.

If $A$ is square and invertible, the unique solution is $\mathbf{x}=A^{-1}\mathbf{b}$. This is efficient for several right-hand sides with the same coefficient matrix.

If $\det A=0$, the inverse method fails. The system may be inconsistent or dependent, so use row reduction, graphs or contextual reasoning to distinguish the cases.

In an application, interpret every component of the solution and reject values that violate physical constraints such as non-negativity.

Three-variable systems use the same structure $A\mathbf{x}=\mathbf b$. On a GDC, inverse, solve or reduced-row-echelon-form commands can obtain the solution.

Row operations preserve the solution set: swap rows, multiply a row by a non-zero scalar, or add a multiple of one row to another.

In reduced row-echelon form, a row $\begin{pmatrix}0&0&0&|&c\end{pmatrix}$ with $c\ne0$ signals inconsistency. A missing pivot with no contradiction signals at least one free variable and infinitely many solutions.

Technology output must be translated back into the original variables and context. A calculator matrix is not a complete written conclusion.

A $2\times2$ matrix transforms a coordinate vector by $\mathbf{x}'=A\mathbf{x}$. The columns of $A$ are the images of the standard basis vectors $\begin{pmatrix}1\\0\end{pmatrix}$ and $\begin{pmatrix}0\\1\end{pmatrix}$.

Common matrices represent stretches, reflections, rotations and shears. A horizontal shear $S=\begin{pmatrix}1&k\\0&1\end{pmatrix}$ maps $(x,y)$ to $(x+ky,y)$.

Composing transformations corresponds to matrix multiplication. With column vectors, $BA\mathbf{x}$ means apply $A$ first and then $B$.

The determinant describes area scaling. If the determinant is zero, a two-dimensional region collapses onto a line or point.

A horizontal shear $S=\begin{pmatrix}1&k\\0&1\end{pmatrix}$, with $k>0$, keeps horizontal levels fixed while shifting points to the right in proportion to their height.

Matrices can turn repeated rules into a compact state model. With a column-state convention, $\mathbf{x}_{n+1}=T\mathbf{x}_n$, so $\mathbf{x}_n=T^n\mathbf{x}_0$.

For a transition matrix acting on probability or population columns, each column records where one current category moves next. Under this convention, non-negative transition proportions in each column sum to 1.

A steady state satisfies $T\mathbf{s}=\mathbf{s}$. It is unchanged by another transition. The total population or total probability supplies an additional equation when needed.

Matrices also model production resources, input-output relationships and weighted totals. Always check dimensions, units, non-negativity and whether the model assumptions remain reasonable over repeated steps.

On an approved GDC, create a matrix with the correct dimensions before entering its elements. Check row and column order directly from the display.

Use matrix commands for addition, multiplication, determinant, inverse, powers and reduced row-echelon form. Menu names vary, so practise the exact workflow on the permitted model.

For a system, either calculate $A^{-1}\mathbf b$, use a simultaneous-equation solver, or row-reduce $[A|\mathbf b]$. Choose a method consistent with the question and show mathematical setup.

Retain full calculator precision. Small displayed decimals near zero may be round-off rather than exact zeros, so use exact inputs whenever possible.

Technology verifies arithmetic but does not replace dimension checks, determinant conditions, model interpretation or an explicit final conclusion.

The most frequent errors are incompatible dimensions, entrywise multiplication, reversing matrix order, missing a negative sign in the inverse, and using an inverse when the determinant is zero.

For every product, write the dimension chain. For every inverse, check the determinant. For every system, define the coefficient matrix, unknown vector and constants vector.

In modelling questions, label state-vector components, state the transition convention and interpret the final vector with units. A mathematically correct array without context is incomplete.

Use exact arithmetic for small matrices when requested. Use technology for larger calculations, but include the mathematical structure and a reasonableness check.

Attempt each question before reading its steps. The set progresses from notation and arithmetic to inverses, systems, transformations and repeated-state models.

Show dimension checks, determinant conditions and contextual interpretation where relevant. Use exact values unless an approximation is requested.

Matrices are reliable when dimensions, multiplication order and invertibility conditions are checked before calculating. In applications, a final matrix must be interpreted as labeled data, a transformation or a state.

Use the formulas below as a quick reference, then practise choosing the correct operation without looking at the section title.