Fudan Advanced Algebra Chapter 1: Determinants §1.1: Second Order Determinants Anki Version Q&A

Question	Answer
Is the number of unknowns in a system of linear equations equal to the number of equations?	It can be equal or not equal.
In the matrix form of a linear equation Ax = b, what does A represent?	Coefficient matrix.
In the matrix form of a linear equation Ax = b, what does x represent?	Unknown vector.
In the matrix form of a linear equation Ax = b, what does b represent?	Constant vector.
What are the solutions to a system of linear equations?	Unique solution, no solution, or infinitely many solutions.
What does the standard form of an n-variable linear equation system look like?	$\begin{array}{c}a_{11} x_{1}+a_{12} x_{2}+\cdots+a_{1 n} x_{n}=b_{1} \\ a_{21} x_{1}+a_{22} x_{2}+\cdots+a_{2 n} x_{n}=b_{2} \\ \cdots \cdots \cdots\\a_{m1} x_{1}+a_{m2} x_{2}+\cdots+a_{m n} x_{n}=b_{m}\end{array}$
$\left\\|\begin{array}{ll}a & b \\ c & d\end{array}\right\\|=$	$ad-bc$
Given a system of linear equations $\left\{\begin{array}{l}a_{11} x_{1}+a_{12} x_{2}=b_{1} \\ a_{21} x_{1}+a_{22} x_{2}=b_{2}\end{array}\right.$ , write down the solution in Cramer's form and verify the derivation?	Multiply both sides of the first equation by $a_{22}$ and both sides of the second equation by $-a_{12}$ , we get: $\left\{\begin{array}{l}a_{11} a_{22} x_{1}+a_{12} a_{22} x_{2}=b_{1} a_{22}, \\-a_{12} a_{21} x_{1}-a_{12} a_{22} x_{2}=-b_{2} a_{12} .\end{array}\right.$ Adding these two equations together, we get: $\left(a_{11} a_{22}-a_{12} a_{21}\right) x_{1}=b_{1} a_{22}-b_{2} a_{12} .$ Therefore, $x_{1}=\frac{b_{1} a_{22}-b_{2} a_{12}}{a_{11} a_{22}-a_{12} a_{21}} .$ By similar method, we can eliminate $x_{1}$ and solve for $x_{2}=\frac{a_{11} b_{2}-a_{21} b_{1}}{a_{11} a_{22}-a_{12} a_{21}} .$
How to remember a system of two linear equations: $\left\{\begin{array}{l}a_{11} x_{1}+a_{12} x_{2}=b_{1} \\ a_{21} x_{1}+a_{22} x_{2}=b_{2}\end{array}\right.$ $x_{1}=\frac{\left\\|\begin{array}{ll}b_{1}&a_{12}\\ b_{2} & a_{22}\end{array}\right\\|}{\left\\|\begin{array}{ll}a_{11}&a_{12}\\ a_{21} & a_{22}\end{array}\right\\|}, x_{2}=\frac{\left\\|\begin{array}{ll}a_{11} & b_{1} \\ a_{21} & b_{2}\end{array}\right\\|}{\left\\|\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right\\|}$	(1) The denominators of x1 and x2 are both the determinant $\left\\|\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right\\|$ , so we only need to arrange the coefficients of the unknowns in the original system in the original order to form a determinant. (2) The first column of the numerator determinant of x1 is the constant column of the original system, and the second column is composed of the coefficients of x2, so this determinant can be seen as replacing the first column of the denominator determinant of x1 and x2 with the constant terms. This rule also applies to the numerator determinant of x2.
How to remember a system of two linear equations: $\left\{\begin{array}{l}a_{11} x_{1}+a_{12} x_{2}=b_{1} \\ a_{21} x_{1}+a_{22} x_{2}=b_{2}\end{array}\right.$$$$x_{1}=\frac{\left\\|\begin{array}{ll}b_{1}&a_{12}\\ b_{2} & a_{22}\end{array}\right\\|}{\left\\|\begin{array}{ll}a_{11}&a_{12}\\ a_{21} & a_{22}\end{array}\right\\|}, x_{2}=\frac{\left\\|\begin{array}{ll}a_{11} & b_{1} \\ a_{21} & b_{2}\end{array}\right\\|}{\left\\|\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right\\|}$	(1) The denominators of x1 and x2 are both the determinant $\left\\|\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right\\|$ , so we only need to arrange the coefficients of the unknowns in the original system in the original order to form a determinant. (2) The first column of the numerator determinant of x1 is the constant column of the original system, and the second column is composed of the coefficients of x2, so this determinant can be seen as replacing the first column of the denominator determinant of x1 and x2 with the constant terms. This rule also applies to the numerator determinant of x2.
What is the value of the determinant $\\|A\\|=\left\\|\begin{array}{cc}a_{11} & a_{12} \\ 0 & a_{22}\end{array}\right\\|$ called?	$a_{11} a_{22}$ upper triangular determinant
What are $a_{11}$ and $a_{22}$ called in a determinant $\\|A\\|=\left\\|\begin{array}{cc}a_{11} & a_{12} \\ 0 & a_{22}\end{array}\right\\|$ ?	Diagonal elements or main diagonal elements
The value of an upper triangular determinant is equal to the product of...	its diagonal elements
If a row or column of a determinant is all zeros, what is the value of the determinant?	0
What is the relationship between the value of a determinant and the value of the determinant obtained by multiplying a row or column of the determinant by a constant c?	It is c times the value of the original determinant. $\\|\begin{array}{cc}\mathrm{ca}_{11} & \mathrm{ca}_{12} \\ \mathrm{a}_{21} & \mathrm{a}_{22}\end{array}\left\\|=\left(\mathrm{ca}_{11}\right) \mathrm{a}_{22}-\left(\mathrm{ca}_{12}\right) \mathrm{a}_{21}=\mathrm{c\\|A}\right\\|$
If two different rows (columns) of a determinant are exchanged, how does the value of the determinant change?	The sign changes
If two rows or two columns of a determinant are proportional (if they are the same, the proportion is 1), what is the value of the determinant?	0
Is the equation $\left\\|\begin{array}{ll}a_{11}+b_{11} & a_{12}+b_{12} \\ a_{21}+b_{21} & a_{22}+b_{22}\end{array}\right\\|=\left\\|\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right\\|+\left\\|\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right\\|$ true? If not, please write the corresponding correct determinant property.	It is not true. The correct form is: $\begin{array}{l}\left\\|\begin{array}{cc}\mathrm{a}_{11} & a_{12} \\ b_{21}+c_{21} & b_{22}+c_{22}\end{array}\right\\|=\left\\|\begin{array}{ll}a_{11} & a_{12} \\ b_{21} & b_{22}\end{array}\right\\|+\left\\|\begin{array}{cc}a_{11} & a_{12} \\ c_{21} & c_{22}\end{array}\right\\| ; \\ \left\\|\begin{array}{ll}b_{11}+c_{11} & a_{12} \\ b_{21}+c_{21} & a_{22}\end{array}\right\\|=\left\\|\begin{array}{cc}b_{11} & a_{12} \\ b_{21} & a_{22}\end{array}\right\\|+\left\\|\begin{array}{cc}c_{11} & a_{12} \\ c_{21} & a_{22}\end{array}\right\\| .\end{array}$
How does the value of a determinant change when a multiple of one row (column) is added to another row (column) of the determinant?	It remains unchanged.
If all elements in a row (column) of a determinant are the sum of two terms, how can the determinant be expressed?	As the sum of two determinants
Given a second-order determinant $\\|A\\|=\left\\|\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right\\|$ , find the transpose of $\\|A\\|$	$\left\\|\begin{array}{ll}a_{11} & a_{21} \\ a_{12} & a_{22}\end{array}\right\\|$
What is the relationship between the value of a determinant and the value of its transpose?	They are the same.