Midterm 1 Review

Solve a system using back-substitution

Start from the last equation in a triangular system and substitute known values upward to solve.

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Solve a system of equations with 2 or 3 variables

Use substitution, elimination, or matrix methods to find solutions for all variables.

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Solve a system by Gaussian Elimination

Transform the augmented matrix to row echelon form (REF) using row operations, then apply back-substitution.

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Find the product of two matrices

Multiply rows by columns:

$$(A\cdot B{)}_{ij}=\sum _k{A}_{ik}{B}_{kj}$$

Only defined when the number of columns in A equals the number of rows in B.

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Find the transpose of a matrix

Swap rows and columns:

$$A_{ij}^T=A_{ji}$$

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Find an elementary matrix

A matrix formed by applying a single row operation to an identity matrix. Used to represent row operations via multiplication.

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Find the inverse of a 2×2 matrix

If:

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$

Then:

$$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$$

Only if $\det (A)\ne 0$.

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Find the inverse of a 3×3 matrix

Two methods:

• Row operations on $A|I\to I|A^{-1}$
• Adjoint method:

$$A^{-1}=\frac{1}{\det (A)}\cdot \text{adj}(A)$$

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Use an inverse matrix to solve a system

Given $AX=B$:

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

Only valid when $A$ is invertible.

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Find the determinant of a 2×2 matrix

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

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Find the determinant of the inverse of a 2×2 matrix

$$\det (A^{-1})=\frac{1}{\det (A)}$$

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Find the determinant of a 3×3 matrix

Use either:

• Cofactor expansion
• Sarrus’ Rule (diagonal method)

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Find minors and cofactors

• Minor of $a_{ij}$: determinant of submatrix excluding row $i$, column $j$
• Cofactor:

$$C_{ij}=(-1{)}^{i+j}\cdot \text{minor}(a_{ij})$$

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Use the definition of a determinant to find an unknown

Set up the determinant expression and solve for the unknown to meet given conditions.

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Identify a matrix as singular or non-singular

• Singular: $det(A)=0$ → no inverse
• Non-singular: $det(A)\ne 0$ → inverse exists

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Solve a system using Cramer’s Rule

For $AX=B$:

$$x_i=\frac{\det (A_i)}{\det (A)}$$

Where $A_i$ is $A$ with its $i$-th column replaced by $B$.

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Use a determinant to find the area of a triangle

For points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$:

$$\text{Area}=\frac{1}{2}\left|\det \left[\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right]\right|$$

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Determine if three points are collinear

Points are collinear if the determinant used in the triangle area formula is zero.

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Matrix addition, subtraction, scalar multiplication

• Addition/Subtraction: element-wise, same dimensions
• Scalar multiplication: each element multiplied by scalar $k$

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Find the adjoint of a matrix

The adjoint is the transpose of the cofactor matrix:

$$\text{adj}(A)=C^T$$

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linear math