Matrix and System of Linear Equations

Practice flashcards reviewing matrix definitions, arithmetic operations, determinants, minors, cofactors, inverses, and Gaussian elimination.

What is a matrix, and how is its order defined?

A matrix is a set of numbers arranged in rows and columns in a rectangular array enclosed by brackets. Its order is defined as $\text{number of rows} \times \text{number of columns}$.

What condition must be met to perform matrix addition or subtraction?

Both matrices must be of the same order.

How is matrix multiplication by a scalar $k$ performed on matrix $M$?

Each element in matrix $M$ is multiplied by the scalar $k$.

What condition must be satisfied for matrix multiplication $AB$ to exist?

The number of columns in matrix $A$ must equal the number of rows in matrix $B$.

How is the determinant of a $2 \times 2$ matrix $A$ with elements $a, b, c, d$ calculated?

$$\det A = |A| = ad - bc$$

What makes a matrix singular, and how does this affect its inverse?

A matrix is singular if its determinant is zero ($|A| = 0$). A singular matrix does not have an inverse.

What is the minor $M_{ij}$ of a matrix $A$?

The minor $M_{ij}$ is the determinant of the matrix obtained by eliminating the $i\text{-th}$ row and $j\text{-th}$ column of matrix $A$.

How is the cofactor $C_{ij}$ calculated from its corresponding minor $M_{ij}$?

$$C_{ij} = (-1)^{i+j} \times M_{ij}$$

What is the formula for calculating the determinant of a $3 \times 3$ matrix $A$ by cofactor expansion along the first row?

$$\det A = a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13}$$

How is the adjugate of a matrix $A$ defined?

The adjugate of matrix $A$ is the transpose of its matrix of cofactors: $\text{Adjugate}(A) = A_{\text{cofactors}}^T$.

What is the formula for finding the inverse of a $3 \times 3$ non-singular matrix $A$?

$$A^{-1} = \frac{1}{\det A} \times A_{\text{cofactors}}^T$$

What three row operations are permitted in Gaussian elimination?

1. Multiply equation $E_i$ by any non-zero constant $\lambda$: $(\lambda E_i) \longrightarrow (E_i)$. 2. Multiply equation $E_j$ by constant $\lambda$ and add to $E_i$: $(E_i + \lambda E_j) \longrightarrow (E_i)$. 3. Transpose equations $E_i$ and $E_j$ in order: $(E_i) \longleftrightarrow (E_j)$.

For matrix $B$ with top row $(7, 6)$ and bottom row $(2, 3)$, what is its determinant?

$$|B| = (7 \times 3) - (6 \times 2) = 21 - 12 = 9$$

For matrix $A$ with row 1 $(2, 1, 4)$, row 2 $(5, 2, 3)$, and row 3 $(8, 7, 3)$, what is the minor $M_{12}$?

$$M_{12} = (5 \times 3) - (3 \times 8) = 15 - 24 = -9$$