“矩阵代数初步”(Introduction to MATRIX ALGEBRA)课程由Prof. A.K.Kaw(University of South Florida)设计并讲授。
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Summary
- Transpose
Let $[A]$ be a $m imes n$ matrix. Then $[B]$ is the transpose of $[A]$ if $b_{ji} = a_{ij}$ for all $i$ and $j$. That is, the $i$-th row and the $j$-th column element of $[A]$ is the $j$-th row and $i$-th column element of $[B]$. Note that $[B]$ is a $n imes m$ matrix and is denoted by $[B] = [A]^{T}$. For example, $$[A] = egin{bmatrix}1& 2& 3\ 4& 5& 6end{bmatrix}Rightarrow [A]^{T} = egin{bmatrix}1& 4\ 2& 5\ 3& 6end{bmatrix}$$ - Symmetric matrix
A square matrix $[A]$ with real elements where $a_{ij} = a_{ji}$ for $i = 1, cdots, n$ and $j = 1, cdots, n$ is called a symmetric matrix. That is, $[A]$ is a symmetric matrix if $[A] = [A]^{T}$. For example, $$[A] = egin{bmatrix}1& 2& 3\ 2& 4& 5\ 3& 5& 7end{bmatrix}$$ - Skew-symmetric matrix
A $n imes n$ matrix is skew-symmetric if $a_{ij} = -a_{ji}$ for $i = 1, cdots, n$ and $j = 1, cdots, n$. That is, $[A]$ is a skew-symmetric matrix if $[A] = -[A]^{T}$. Note that the diagonal elements must be zero in a skew-symmetric matrix. For example, $$[A] = egin{bmatrix}0& 2& 3\ -2& 0& 5\ -3& -5& 0end{bmatrix}$$ - Trace of matrix
The trace of a $n imes n$ matrix $[A]$ is the sum of the diagonal entries of $[A]$, that is, $$ ext{tr}[A] = sum_{i=1}^{n}a_{ii}$$ For example, $$[A] = egin{bmatrix}1& 2& 3\ 2& 4& 5\ 3& 5& 7end{bmatrix}Rightarrow ext{tr}[A] = 1 + 4 +7=12$$ - Determinant
Let $[A]$ be a $n imes n$ matrix.- The minor of entry $a_{ij}$ is denoted by $M_{ij}$ and is defined as the determinant of the $(n-1) imes(n-1)$ sub-matrix of $[A]$, where the sub-matrix is obtained by deleting the $i$-th row and $j$-th column of the matrix $[A]$. The determinant is then given by $$det(A) = sum_{j=1}^{n}(-1)^{i+j}a_{ij}M_{ij}, ext{for any} i=1, 2, cdots, n$$ or $$det(A) = sum_{i=1}^{n}(-1)^{i+j}a_{ij}M_{ij}, ext{for any} j=1, 2, cdots, n$$ For example, $$[A] = egin{bmatrix}1& 2& 3\ 2& 4& 5\ 3& 5& 7end{bmatrix}$$ $$Rightarrow det(A) =(-1)^{1+1}cdot1cdotegin{vmatrix}4& 5 \ 5& 7end{vmatrix} + (-1)^{1+2}cdot2cdotegin{vmatrix}2& 5 \ 3& 7end{vmatrix} + (-1)^{1+3}cdot3cdotegin{vmatrix}2& 4 \ 3& 5end{vmatrix}$$ $$=(4 imes7-5 imes5) -2 imes(2 imes7-3 imes5) + 3 imes(2 imes5 - 3 imes4) = -1$$ Note that for a $2 imes2$ matrix $[A] = egin{bmatrix}a& b\ c& dend{bmatrix}$, $det(A) = ad-bc$.
- The number $(-1)^{i+j}M_{ij}$ is called the cofactor of $a_{ij}$ and is denoted by $C_{ij}$. The formula for the determinant can then be written as $$det(A) = sum_{j=1}^{n}a_{ij}C_{ij}, ext{for any} i=1, 2, cdots, n$$ or $$det(A) = sum_{i=1}^{n}a_{ij}C_{ij}, ext{for any} j=1, 2, cdots, n$$
- If $[A]$ and $[B]$ are square matrices of same size, then $$det(Acdot B) = det(A)cdotdet(B)$$
- $det(A) = 0$ if
- A row or a column is zero, or
- A row (column) is proportional to another row (column).
- If a row (column) is multiplied by $k$ to result in matrix $[B]$, then $$det(B) = kcdotdet(A)$$
- If $[B] = kcdot[A]$, then $$det(B)=k^{n}det(A)$$
- If $[A]$ is a $n imes n$ upper or lower triangular matrix, then $$det(A) = prod_{i=1}^{n}a_{ii}$$
- If $[B]$ is row-equivalent to $[A]$, then $$egin{cases} R_ileftrightarrow R_j: & det(B) = -det(A);\ tR_i: & det(B) = tdet(A);\ R_i ightarrow R_i+tR_j: &det(B) = det(A).end{cases}$$
Selected Problems
1. Let $$[A] = egin{bmatrix}25& 3& 6\ 7& 9& 2end{bmatrix}$$ Find $[A]^{T}$.
Solution:
$$[A]^{T} = egin{bmatrix}25& 7\ 3& 9\ 6& 2end{bmatrix}$$
2. If $[A]$ and $[B]$ are two $n imes n$ symmetric matrices, show that $[A]+[B]$ is also symmetric.
Solution:
Let $[C]=[A]+[B]$, so we have $$c_{ij} = a_{ij} + b_{ij} = a_{ji} + b_{ji} =c_{ji}$$ that is, $[C]=[C]^{T}$.
3. What is the trace of $$[A] = egin{bmatrix}7& 2& 3& 4\ -5& -5& -5& -5\ 6& 6& 7& 9\ -5& 2& 3& 10end{bmatrix}$$
Solution:
$$ ext{tr}[A] = 7-5+7+10=19$$
4. Find the determinant of $$[A] = egin{bmatrix}10& -7& 0\ -3& 2.099& 6\ 5& -1& 5end{bmatrix}$$
Solution:
$$det(A)=(-1)^{1+1} imes10 imesegin{vmatrix}2.099& 6\ -1& 5end{vmatrix} + (-1)^{1+2} imes(-7) imesegin{vmatrix}-3& 6\ 5& 5end{vmatrix}$$ $$=10 imes(2.099 imes5+1 imes6) + 7 imes(-15-30) = -150.05$$
5. What is the value of a $n imes n$ matrix $det(3[A])$?
Solution:
$$det(3[A]) = 3^ndet(A)$$
6. For a $5 imes5$ matrix $[A]$, the first row is interchanged with the fifth row, what is the determinant of the resulting matrix $[B]$?
Solution:
The sign would be changed if interchaged two row (column). Thus $$det(B) = -det(A)$$
7. What is the determinant of $$[A] = egin{bmatrix}0& 1& 0& 0\ 0& 0& 1& 0\ 0& 0& 0& 1\ 1& 0& 0& 0end{bmatrix}$$
Solution:
$$[A] = egin{bmatrix}0& 1& 0& 0\ 0& 0& 1& 0\ 0& 0& 0& 1\ 1& 0& 0& 0end{bmatrix}Rightarrow R_1leftrightarrow R_4 egin{bmatrix}1& 0& 0& 0\ 0& 0& 1& 0\ 0& 0& 0& 1\ 0& 1& 0& 0end{bmatrix}$$ $$Rightarrow R_2leftrightarrow R_3 egin{bmatrix}1& 0& 0& 0\ 0& 0& 0& 1\ 0& 0& 1& 0\0& 1& 0& 0end{bmatrix}$$ $$Rightarrow R_2leftrightarrow R_4 egin{bmatrix}1& 0& 0& 0\0& 1& 0& 0\ 0& 0& 1& 0\0& 0& 0& 1end{bmatrix}=[B]$$ Thus $det(A) = (-1)^{3}det(B)=-1$.
8. Find the determinant of $$[A]=egin{bmatrix}0& 0& 0\ 2& 3& 5\ 6& 9& 2end{bmatrix}$$
Solution:
$det(A)=0$ since the first row is zero.
9. Find the determinant of $$[A]=egin{bmatrix}0& 0& 2& 3\ 0& 2& 3& 5\ 6& 7& 2& 3\ 6.6& 7.7& 2.2& 3.3end{bmatrix}$$
Solution:
Since $R_4 = 1.1R_3$, so $det(A) = 0$.
10. Find the determinant of $$[A]=egin{bmatrix}5& 0& 0& 0\ 0& 3& 0& 0\ 2& 5& 6& 0\ 1& 2& 3& 9end{bmatrix}$$
Solution:
This is a lower triangular matrix and hence $$det(A) = 5 imes3 imes6 imes9=810$$
11. Given the matrix $$[A]=egin{bmatrix}125& 25& 5& 1\ 512& 64& 8& 1\ 1157& 89& 13& 1\ 8& 4& 2& 1end{bmatrix}$$ and $det(A) = -32400$. Find the determinant of $$[A_1]=egin{bmatrix}125& 25& 5& 1\ 512& 64& 8& 1\ 1141& 81& 9& -1\ 8& 4& 2& 1end{bmatrix};$$ $$[A_2]=egin{bmatrix}125& 25& 1& 5\ 512& 64& 1& 8\ 1157& 89& 1& 13\ 8& 4& 1& 2end{bmatrix};$$ $$[A_3] = egin{bmatrix} 125& 25& 5& 1\ 1157& 89& 13& 1\ 512& 64& 8& 1\8& 4& 2& 1end{bmatrix};$$ $$[A_4] = egin{bmatrix} 125& 25& 5& 1\ 1157& 89& 13& 1\ 8& 4& 2& 1\ 512& 64& 8& 1end{bmatrix};$$ $$[A_5] = egin{bmatrix} 125& 25& 5& 1\ 512& 64& 8& 1\ 1157& 89& 13& 1\ 16& 8& 4& 2 end{bmatrix}.$$
Solution:
$$[A]=egin{bmatrix}125& 25& 5& 1\ 512& 64& 8& 1\ 1157& 89& 13& 1\ 8& 4& 2& 1end{bmatrix}Rightarrow R_3-2R_4 egin{bmatrix}125& 25& 5& 1\ 512& 64& 8& 1\ 1141& 81& 9& -1\ 8& 4& 2& 1end{bmatrix}=[A_1]$$ Thus $det(A_1) = det(A) =-32400$. $$[A]=egin{bmatrix}125& 25& 5& 1\ 512& 64& 8& 1\ 1157& 89& 13& 1\ 8& 4& 2& 1end{bmatrix}Rightarrow C_3leftrightarrow C_4 egin{bmatrix}125& 25& 1& 5\ 512& 64& 1& 8\ 1157& 89& 1& 13\ 8& 4& 1& 2end{bmatrix} = [A_2]$$ Thus $det(A_2)=-det(A)=32400$. $$[A]=egin{bmatrix}125& 25& 5& 1\ 512& 64& 8& 1\ 1157& 89& 13& 1\ 8& 4& 2& 1end{bmatrix}Rightarrow R_2leftrightarrow R_3egin{bmatrix} 125& 25& 5& 1\ 1157& 89& 13& 1\ 512& 64& 8& 1\8& 4& 2& 1end{bmatrix}= [A_3]$$ Thus $det(A_3) = -det(A) = 32400$. $$[A]=egin{bmatrix}125& 25& 5& 1\ 512& 64& 8& 1\ 1157& 89& 13& 1\ 8& 4& 2& 1end{bmatrix}Rightarrow egin{cases} R_2leftrightarrow R_3\ R'_3leftrightarrow R_4end{cases} egin{bmatrix} 125& 25& 5& 1\ 1157& 89& 13& 1\ 8& 4& 2& 1\ 512& 64& 8& 1end{bmatrix} = [A_4]$$ Thus $det(A_4) = (-1)^2det(A) = -32400$. $$[A]=egin{bmatrix}125& 25& 5& 1\ 512& 64& 8& 1\ 1157& 89& 13& 1\ 8& 4& 2& 1end{bmatrix} Rightarrow 2R_4egin{bmatrix} 125& 25& 5& 1\ 512& 64& 8& 1\ 1157& 89& 13& 1\ 16& 8& 4& 2 end{bmatrix} = [A_5]$$ Thus $det(A_5) = 2det(A) = -64800$.
12. Find the determinant of $$[A] = egin{bmatrix}25& 5& 1\ 64& 8& 1\ 144& 12& 5end{bmatrix}$$
Solution:
$$det(A) = (-1)^{1+3}a_{13}M_{13}+(-1)^{2+3}a_{23}M_{23} + (-1)^{3+3}a_{33}M_{33}$$ $$ = egin{vmatrix}64& 8\ 144& 12end{vmatrix} - egin{vmatrix}25& 5\ 144& 12end{vmatrix} + 5 imes egin{vmatrix}25& 5\ 64& 8end{vmatrix} = -564$$
13. Show that if $[A][B]=[I]$, where $[A]$, $[B]$ and $[I]$ are matrices of $n imes n$ size and $[I]$ is an identity matrix, then $det(A) eq0$ and $det(B) eq0$.
Solution: $$det(A)det(B)=det(AB) =det(I) = 1$$ $$Rightarrow det(A) eq0, det(B) eq0.$$
14. If the determinant of a $4 imes4$ matrix $[A]$ is given as 20, then what is the determinant of $5[A]$?
Solution:
$$det(k[A])=k^ndet(A)$$ $$Rightarrow det(5[A]) = 5^4det(A) = 625 imes20=12500$$
15. If the matrix product $[A][B][B]$ is defined, what is $([A][B][C])^{T}$?
Solution:
$$([A][B])^{T}=[B]^{T}[A]^{T}$$ $$Rightarrow ([A][B][C])^{T}=[C]^{T}([A][B])^{T}=[C]^{T}[B]^{T}[A]^{T}$$
16. The determinant of the matrix $$[A] = egin{bmatrix}25& 5& 1\ 0& 3& 8\ 0& 9& aend{bmatrix}$$ is 50. What is the value of $a$?
Solution:
$$det(A) = 25 imesegin{vmatrix}3& 8\ 9& aend{vmatrix} = 25 imes(3a-72)=50$$ $$Rightarrow a={74over3}$$
17. $[A]$ is a $5 imes 5$ matrix and a matrix $[B]$ is obtained by the row operations of replacing Row1 with Row3, and then Row3 is replaced by a linear combination of $2 imes$Row3$+4 imes$Row2. If $det(A)=17$, then what is the value of $det(B)$?
Solution:
The process is $$[A]Rightarrow R_1leftrightarrow R_3 Rightarrow 2R_3Rightarrow R_3+4R_2Rightarrow [B]$$ Thus $$det(B) = (-1) imes2cdotdet(A) = -34$$