Matrix Algebra Flashcards

Practice flashcards covering definitions, properties, operations, determinants, inverses, and system-solving methods in Matrix Algebra.

How is a matrix defined?

A matrix is a set of $n \times m$ real numbers placed in a rectangular table of $n$ lines (rows) and $m$ columns, where each element is located at the intersection of the $i$-th line and $j$-th column for every $1 \le i \le n$ and $1 \le j \le m$.

What is the condition for two matrices $A_n^m$ and $B_n^m$ to be equal?

Two matrices $A_n^m$ and $B_n^m$ are equal if $a_i^j = b_i^j$ for all $1 \le i \le n$ and $1 \le j \le m$.

How is the transpose of a matrix $A_n^m$ defined?

The transpose of $A_n^m$, designated as $A^t$, is a matrix of $m$ lines and $n$ columns formed by taking the elements by line and writing them by column, such that $(a_i^j)_{A^t} = (a_j^i)_A$.

What is a line matrix?

A line matrix is a matrix consisting of $1$ line and $m$ columns, denoted as $M_1^m = (a_1^1 \quad a_1^2 \quad \dots \quad a_1^m)$.

What is a column matrix?

A column matrix is a matrix consisting of $n$ lines and $1$ column, denoted as $M_n^1$.

What defines a zero (null) matrix?

A zero matrix is a matrix where all entries are equal to zero ($a_i^j = 0$ for every location $(i, j)$).

What defines a square matrix of order $n$?

A square matrix is a matrix where the number of lines equals the number of columns ($n = m$), denoted as $M_n^n$.

How is the diagonal of a square matrix defined?

The diagonal of a square matrix $M_n^n$ is the set of elements $a_i^i$ (where line index equals column index).

What is an upper triangular matrix?

An upper triangular matrix is a square matrix where all the elements under the main diagonal are equal to zero ($a_i^j = 0$ for $i > j$).

What is a lower triangular matrix?

A lower triangular matrix is a square matrix where all the elements above the main diagonal are equal to zero ($a_i^j = 0$ for $i < j$).

What is a diagonal matrix?

A diagonal matrix is a square matrix where all elements outside the main diagonal are equal to zero ($a_i^j = 0$ for $i \neq j$).

What is an identity matrix $I$?

An identity matrix is a diagonal matrix where all the elements on the diagonal are equal to $1$ ($a_i^i = 1$ and $a_i^j = 0$ for $i \neq j$).

What is a symmetric matrix?

A symmetric matrix is a square matrix where all elements located symmetrically with respect to the diagonal are equal ($a_i^j = a_j^i$, or $A^t = A$).

What is an anti-symmetric matrix?

An anti-symmetric matrix is a square matrix where all elements located symmetrically with respect to the diagonal are opposite ($a_i^j = -a_j^i$, or $A^t = -A$).

How is matrix addition defined for $A_n^m$ and $B_n^m$?

The sum $C_n^m = A_n^m + B_n^m$ is the matrix where each element $c_i^j = a_i^j + b_i^j$ for all $1 \le i \le n$ and $1 \le j \le m$.

What are the main algebraic properties of matrix addition?

Matrix addition is commutative ($A + B = B + A$), associative ($A + (B + C) = (A + B) + C$), has the zero matrix as a neutral element ($A + 0 = A$), and satisfies $(A + B)^t = A^t + B^t$.

How is scalar multiplication defined for a matrix $A$ and scalar $\alpha$?

To multiply a matrix by a scalar $\alpha$ is to multiply every entry of the matrix by $\alpha$.

What dimension condition must be satisfied to multiply matrix $A_n^p$ by matrix $B_p^m$?

The number of columns of matrix $A$ ($p$) must equal the number of lines of matrix $B$ ($p$), yielding a product matrix $C_n^m$.

What formula gives each entry $c_i^j$ in matrix multiplication $C_n^m = A_n^p \times B_p^m$?

$$c_i^j = \sum_{k=1}^p a_i^k b_k^j$$

What is the transpose of a product of two matrices $(A \times B)^t$?

The transpose of the product of two matrices equals the product of their transposes in reverse order: $(A \times B)^t = B^t \times A^t$.

Do zero divisors exist in matrix multiplication?

Yes, two non-zero matrices $A$ and $B$ can produce a zero product ($A \times B = 0$).

Is cancellation allowed in matrix multiplication (does $A \times B = A \times C$ imply $B = C$)?

No, in matrix multiplication you cannot simplify/cancel; $A \times B = A \times C$ does not imply $B = C$.

What is the transpose of an upper triangular matrix?

The transpose of an upper triangular matrix is a lower triangular matrix (and vice versa).

How is the determinant of a $2 \times 2$ square matrix $A = \begin{pmatrix} a_1^1 & a_1^2 \\ a_2^1 & a_2^2 \end{pmatrix}$ calculated?

$$|A| = a_1^1 a_2^2 - a_1^2 a_2^1$$

How are sign factors determined when expanding a determinant across a row or column?

The sign factor for element $a_i^j$ is calculated using $(-1)^{i+j}$.

What is the determinant of a matrix if an entire row or column consists of zeros?

The determinant is equal to zero ($|A| = 0$).

What is the determinant of a matrix if two rows or two columns are proportional?

The determinant is equal to zero ($|A| = 0$).

How does multiplying all entries of an order $n$ square matrix by scalar $\alpha$ affect its determinant?

The determinant is multiplied by $\alpha^n$, giving $|\alpha A| = \alpha^n |A|$.

How does the determinant of $A^t$ compare to the determinant of $A$?

The determinant of the transpose of a matrix is equal to the determinant of the initial matrix ($|A^t| = |A|$).

How is the determinant of a diagonal or triangular matrix calculated?

It is equal to the product of all the elements on the main diagonal.

What happens to the sign of a determinant when two lines or two columns are swapped?

The determinant is multiplied by $-1$.

What condition defines the inverse matrix $A^{-1}$ of a square matrix $A$?

$A \times A^{-1} = A^{-1} \times A = I$, where $I$ is the identity matrix.

What is the existence condition for the inverse of a square matrix $A$?

The determinant of $A$ must be non-zero ($|A| \neq 0$).

What is the inverse of a matrix product $(A \times B)^{-1}$?

$$(A \times B)^{-1} = B^{-1} \times A^{-1}$$

What are the 5 steps to compute the inverse of a matrix using the co-factors method?

1. Existence condition ($|A| \neq 0$).
2. Pre-co-matrix $A'$ (minors).
3. Co-matrix $A''$ (apply signs $(-1)^{i+j}$).
4. Adjoint matrix A^* = (A'')^t$.\n5. Inverse matrix A^{-1} = \frac{1}{|A|} A^*$$.

How is a linear system of $n$ equations with $p$ unknowns expressed in matrix notation?

$A_{(n;p)} \cdot X_{(p;1)} = B_{(n;1)}$, where $A$ is the coefficient matrix, $X$ is the matrix of unknowns, and $B$ is the second member matrix.

How is a matrix system $A \cdot X = B$ solved using the Inverse Matrix Method?

If $|A| \neq 0$, the inverse $A^{-1}$ exists and multiplying both sides gives $X = A^{-1} B$.

What is Cramer's formula for finding unknown $X_i$ in a matrix system $A \cdot X = B$?

$X_i = \frac{\Delta_{X_i}}{\Delta}$, where $\Delta = |A| \neq 0$ and $\Delta_{X_i}$ is the determinant of matrix $A$ with column $i$ replaced by the second member matrix $B$.

How does the Gauss Elimination Method solve a matrix system $A X = B$?

Row operations are applied to the augmented matrix $(A|B)$ to transform $A$ into an upper triangular matrix $A'$ with $1$s on the diagonal, giving $A' X = B'$, which is then solved by back-substitution.