Algebra

The One Thing

Matrices and determinants provide systematic methods for representing data and solving simultaneous linear equations. Their use requires dimensionally valid operations, nonzero determinants where inverses or Cramer’s Rule are applied, and interpretation of solutions within the original context.

Definitions and Results

• Matrix: A rectangular arrangement of numbers or expressions in rows and columns. A matrix $A=[a_{ij}]$ of order $m\times n$ has $m$ rows and $n$ columns.
• Order of a Matrix: The number of rows multiplied by the number of columns, written $m\times n$.
• Row Matrix: A matrix having only one row.
• Column Matrix: A matrix having only one column.
• Square Matrix: A matrix with the same number of rows and columns.
• Zero Matrix: A matrix in which every element is zero.
• Identity Matrix: A square matrix with 1s on the main diagonal and 0s elsewhere; it acts as the multiplicative identity.
• Transpose: The matrix obtained by interchanging rows and columns, denoted $A^T$. The rules are

$$(A^T{)}^T=A,(A+B{)}^T=A^T+B^T,(AB{)}^T=B^T{A}^T.$$

• Matrix Addition: The addition of corresponding elements of matrices having the same order. Matrices can be added or subtracted only when their orders are identical.
• Matrix Multiplication: For $A$ of order $m\times n$ and $B$ of order $n\times p$, $AB$ is defined and has order $m\times p$. Matrix multiplication is generally not commutative, so $AB$ may not equal $BA$.
• Determinant: A scalar associated with a square matrix, useful for testing solvability and finding inverses. For

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],\det (A)=ad-bc.$$

• Minor: The determinant obtained after deleting the row and column containing a particular element.
• Cofactor: The signed minor of an element:

$$C_{ij}=(-1{)}^{i+j}{M}_{ij}.$$

• Adjoint: The transpose of the cofactor matrix of a square matrix.
• Inverse Matrix: For a nonsingular matrix $A$, $A^{-1}$ satisfies

$$A{A}^{-1}=A^{-1}A=I.$$ For a square matrix, $$A^{-1}=\frac{\mathrm{adj}(A)}{|A|},$$ provided $|A|\ne 0$.

• Singular Matrix: A square matrix whose determinant is zero.
• Nonsingular Matrix: A square matrix whose determinant is nonzero.
• Invertibility: A matrix is invertible if and only if its determinant is nonzero.
• System of Linear Equations: Two or more linear equations involving common variables that must be satisfied simultaneously.
• Unique Solution: A system has exactly one solution, commonly when the coefficient determinant is nonzero.
• No Solution: A system is inconsistent because no values satisfy all equations simultaneously.
• Infinitely Many Solutions: A system is dependent, so multiple values satisfy all equations.
• Cramer’s Rule: A determinant-based method for solving a system of linear equations when the coefficient determinant is nonzero.
• Matrix Method: A method that expresses a system as $AX=B$ and solves it using an inverse matrix when $A^{-1}$ exists. If $A$ is invertible,

$$X=A^{-1}B.$$

• Mathematical Modelling: The process of translating a real-life situation into algebraic equations and interpreting the solution.
• Determinant Properties: For a $3\times 3$ determinant, expansion may be carried out along any row or column using elements and their cofactors. Interchanging two rows or two columns changes the sign. Identical or proportional rows or columns give a determinant of zero. Multiplying one row or column by a constant multiplies the determinant by that constant.

Worked Methods

Matrix operations

• Identify the order of each matrix.
• Add or subtract corresponding elements only if the matrices have the same order.
• For multiplication, check that the number of columns of the first matrix equals the number of rows of the second.
• If $A$ is $m\times n$ and $B$ is $n\times p$, calculate the row-by-column products to obtain an $m\times p$ matrix.
• Do not assume that $AB=BA$; matrix multiplication is generally not commutative.
• For a transpose, interchange rows and columns and apply

$$(A^T{)}^T=A,(A+B{)}^T=A^T+B^T,(AB{)}^T=B^T{A}^T.$$

Determinants, minors, and cofactors

• For a $2\times 2$ matrix

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$$ calculate $$\det (A)=ad-bc.$$

• For a larger square matrix, select a row or column for expansion.
• For each selected element, delete its row and column to obtain its minor $M_{ij}$.
• Apply the sign rule

$$C_{ij}=(-1{)}^{i+j}{M}_{ij}$$ to obtain the cofactor.

• Multiply each selected element by its cofactor and add the results.
• Use determinant properties to simplify calculations where applicable: interchanging rows or columns changes the sign; identical or proportional rows or columns give zero; multiplying a row or column by a constant multiplies the determinant by that constant.

Finding an inverse matrix

• Confirm that the matrix is square.
• Calculate its determinant $|A|$.
• If $|A|=0$, the matrix is singular and has no inverse.
• If $|A|\ne 0$, calculate each minor and cofactor.
• Form the cofactor matrix and transpose it to obtain $\mathrm{adj}(A)$.
• Use

$$A^{-1}=\frac{\mathrm{adj}(A)}{|A|}.$$

• Verify, where required, that

$$A{A}^{-1}=A^{-1}A=I.$$

Solving a system using the matrix method

• Write the simultaneous equations in the form

$$AX=B.$$

• Identify the coefficient matrix $A$, the variable matrix $X$, and the constants matrix $B$.
• Determine whether $|A|\ne 0$, so that $A^{-1}$ exists.
• Multiply by $A^{-1}$:

$$A^{-1}AX=A^{-1}B.$$

• Use $A^{-1}A=I$ to obtain

$$X=A^{-1}B.$$

• Check the values by substitution into the original equations.

Solving two equations using Cramer’s Rule

For $$a_{1}x+b_{1}y=c_1,a_{2}x+b_{2}y=c_2:$$

• Calculate the coefficient determinant:

$$D=a_1{b}_2-a_2{b}_1.$$

• If $D=0$, the standard form of Cramer’s Rule does not apply; the system may have no solution or infinitely many solutions.
• Form $D_x$ by replacing the $x$-coefficient column with the constants column.
• Form $D_y$ by replacing the $y$-coefficient column with the constants column.
• Provided $D\ne 0$, calculate

$$x=\frac{D_x}{D},y=\frac{D_y}{D}.$$

• Check the result by substitution.

Examining three linear equations

• Form the coefficient determinant from the coefficients of the three variables.
• Form the determinants obtained by replacing each coefficient column in turn with the constants column.
• Compare the coefficient determinant with these replacement determinants.
• Use the comparison to examine whether the system is consistent and whether it has a unique solution, no solution, or infinitely many solutions.

Mathematical modelling in applications

• Define variables clearly.
• Translate the given conditions into algebraic equations.
• Use matrix methods, determinants, Cramer’s Rule, or ordinary algebra to solve the equations.
• Check the result by substitution, estimation, or dimensional reasoning.
• Interpret the solution in the original situation.
• Apply practical restrictions: numbers of products, people, or units are usually nonnegative and may need to be whole numbers.
• Keep units consistent in applications involving cost, revenue, production, and resource allocation.

Where It Goes Wrong

• Adding or subtracting matrices of different orders; corresponding-element operations require the matrices to have the same order.
• Multiplying matrices without checking compatibility; for $A$ of order $m\times n$ and $B$ of order $n\times p$, the product has order $m\times p$.
• Treating matrix multiplication as commutative and assuming $AB=BA$.
• Applying $A^{-1}=\mathrm{adj}(A)/|A|$ or Cramer’s Rule without first checking that the determinant is nonzero.
• Losing the cofactor sign in $C_{ij}=(-1{)}^{i+j}{M}_{ij}$, or forgetting that interchanging two rows or columns changes the determinant’s sign.
• Accepting an algebraic answer without checking substitution, units, nonnegative values, whole-number restrictions, or whether it is meaningful in the original context.

What Gets Asked

• Define and identify matrices by type and order, including row, column, square, zero, and identity matrices.
• Perform matrix addition, subtraction, multiplication, and transposition, including the transpose rules.
• Calculate $2\times 2$ and $3\times 3$ determinants using minors, cofactors, expansion, and determinant properties.
• Determine whether a matrix is singular, nonsingular, or invertible.
• Find the adjoint and inverse of a square matrix.
• Solve simultaneous linear equations using Cramer’s Rule, including

$$D=a_1{b}_2-a_2{b}_1,x=\frac{D_x}{D},y=\frac{D_y}{D}.$$

• Express and solve systems in the form $AX=B$, using $X=A^{-1}B$.
• Classify systems as having a unique solution, no solution, or infinitely many solutions.
• Examine consistency in three-equation systems by comparing the coefficient determinant with determinants formed by replacing columns with constants.
• Form and solve algebraic models for business planning, costing, production, decision-making, and resource allocation.
• Interpret results using practical restrictions, consistent units, substitution, estimation, and dimensional reasoning.