Fudan Advanced Algebra Chapter 1: Determinants §1.1: Second Order Determinants Anki version Q&A test question

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, $x_{1}=\frac{b_{1} a_{22}-b_{2} a_{12}}{a_{11} a_{22}-a_{12} a_{21}} .$ Using a similar method to eliminate $x_{1}$ , we get: $x_{2}=\frac{a_{11} b_{2}-a_{21} b_{1}}{a_{11} a_{22}-a_{12} a_{21}} .$
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 just 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. The same rule 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 just 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. The same rule 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?	The upper triangular determinant is equal to the product of its diagonal elements.
What are $a_{11}$ and $a_{22}$ called in the 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 one row or one 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 one row or one 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.	No, 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 one row (column) is multiplied by a constant and added to another row (column)?	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.