matrix-determinants

Types of Matrices

1. Row Matrix: A matrix with only 1 row.
2. Column Matrix: A matrix with only 1 column.
3. Square Matrix: A matrix with an equal number of rows and columns.
4. Rectangular Matrix: A matrix with an unequal number of rows and columns.
5. Zero Matrix: A matrix where all elements are zero, regardless of size.
6. Diagonal Matrix: A square matrix where all elements except for the main diagonal are zero.
7. Scalar Matrix: A square matrix with a constant value on the diagonal and zeros elsewhere.
8. Unity Matrix: A matrix where all elements are 1, regardless of size.
9. Identity Matrix: A square matrix with 1s along the main diagonal and 0s elsewhere.
10. Upper Triangular Matrix: A square matrix where all elements below the diagonal are zero.
11. Lower Triangular Matrix: A square matrix where all elements above the diagonal are zero.
12. Symmetric Matrix: A square matrix that is equal to its transpose.
13. Skew-Symmetric Matrix: A square matrix that is equal to the negative of its transpose.
14. Non-Singular (Invertible) Matrix: A square matrix with a non-zero determinant.
15. Singular (Non-Invertible) Matrix: A square matrix with a determinant of zero.
16. Involutory Matrix: A square matrix that is its own inverse, meaning $A^2=I$, where $I$ is the identity matrix.
17. Orthogonal Matrix: A square matrix such that $A{A}^T=I$.
    • Important: Its determinant is $\pm 1$.
    • Conditions:
      • Magnitude of every column must be 1.
      • Dot product of columns must be zero.
18. Idempotent Matrix: A square matrix such that $A^n=A$ for every $n\ge 2$.
19. Matrix with $A^2=I$: Its determinant is either 1 or -1.

Key Points about Matrices

1. Matrix operations always yield matrices as results.
2. Left Distribution Law: $A(B+C)=AB+AC$.
3. Right Distribution Law: $(B+C)A=BA+CA$.
4. If $A+A^T$ is symmetric, then $A-A^T$ is skew-symmetric.
5. Any square matrix can be expressed as a sum of a symmetric and skew-symmetric matrix: $A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)$
   • Here, $P=\frac{1}{2}(A+A^T)$ (symmetric) and $Q=\frac{1}{2}(A-A^T)$ (skew-symmetric).
6. Transpose properties:
   • $(AB{)}^T=B^T{A}^T$ and $(ABC{)}^T=C^T{B}^T{A}^T$.
7. The zero matrix is the only matrix that is both symmetric and skew-symmetric.
8. The product $AB$ is symmetric if and only if $AB=BA$.
9. The sum $AB+BA$ is symmetric, while $AB-BA$ is skew-symmetric.
10. $(A^T{)}^n=(A^n{)}^T$.

Properties of Determinants

1. The value of a determinant remains unchanged if its rows and columns are interchanged.
2. If any two rows (or columns) of a determinant are interchanged, the sign of the determinant changes.
3. If any two rows (or columns) of a determinant are identical, the value of the determinant is zero.
4. If all elements of a row (or column) are multiplied by a scalar $K$, the value of the determinant is also multiplied by $K$: $K\times \left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=\left|\begin{array}{ccc}K{a}_1& K{b}_1& K{c}_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$
5. If some or all elements of a row (or column) are expressed as the sum of two (or more) terms, the determinant can be expressed as the sum of two (or more) determinants of the same order: $\left|\begin{array}{ccc}{a}_1+x& b_1+y& c_1+z\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|+\left|\begin{array}{ccc}x& y& z\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$
6. The value of the determinant is not altered by adding or subtracting a multiple of any row (or column) to another row (or column): $\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=\left|\begin{array}{ccc}{a}_1+K{a}_3& b_1+K{b}_3& c_1+K{c}_3\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$
7. Factor Theorem: If substituting $x=a$ makes the determinant vanish, then $(x-a)$ is a factor of the determinant.

Cofactor Determinant

The determinant formed by replacing all elements of a determinant with their respective cofactors is called the cofactor determinant. If $|A{|}_C$ is the cofactor determinant of $A$, then: $|A|\_C=|A{|}^{n-1}$

Determinant Results

1. $|AB|=|A|\times |B|$.
2. $(A^{-1}{)}^{-1}=A$
3. $(AB{)}^{-1}=B^{-1}{A}^{-1}$
4. $(A^T{)}^{-1}=(A^{-1}{)}^T$
5. $|\text{adj}(A)|=|A{|}^{n-1}$
6. $|A^{-1}|=|A{|}^{-1}$
7. $\text{adj}(AB)=\text{adj}(B)\cdot \text{adj}(A)$
8. $\text{adj}(A^m)=\text{adj}(A{)}^m$
9. $\text{adj}(kA)=k^{n-1}\text{adj}(A)$
10. $\text{adj}(A^T)=(\text{adj}(A){)}^T$
11. $\text{adj}(\text{adj}(A))=|A{|}^{n-2}A$
12. $\text{adj}(\text{adj}(A))|=|A{|}^{(n-1{)}^2}$

Determinants Key Points

1. Determinant operations always yield scalar values.
2. If $k$ is any scalar and $A$ is any matrix, then $|kA|=k^n|A|$, where $n$ is the order of matrix $A$.
3. The area of a triangle with vertices $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$ can be found using the determinant: $\text{Area}=\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$
4. The inverse of a square matrix $A$ (if it exists) is given by: $A^{-1}=\frac{\text{adj}(A)}{|A|}\text{if }|A|\ne 0$
5. The product of a matrix and its adjugate is: $A\cdot \text{adj}(A)=\text{adj}(A)\cdot A=|A|I$
6. The product of a matrix and its inverse is: $A{A}^{-1}=I=A^{-1}A$
7. The inverse $A^{-1}$ exists only when $A$ is a non-singular matrix.
8. The inverse of a square matrix, if it exists, is unique.
9. If $A$ is a non-singular matrix, then both $A^{-1}$ and $A^T$ are also non-singular.
10. The determinant of a skew-symmetric matrix of odd order is zero.
11. The determinant of a skew-symmetric matrix of even order is a non-zero perfect square.
12. For a square matrix A, $|A^T|=|A|$.
13. A square matrix B is called the inverse of A if $AB=BA=I$

System of Linear Equations

For a system of linear equations with three variables: $a_{1}x+b_{1}y+c_{1}z=d_1$ $a_{2}x+b_{2}y+c_{2}z=d_2$ $a_{3}x+b_{3}y+c_{3}z=d_3$

$D=\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$ $D_x=\left|\begin{array}{ccc}{d}_1& b_1& c_1\\ d_2& b_2& c_2\\ d_3& b_3& c_3\end{array}\right|D_y=\left|\begin{array}{ccc}{a}_1& d_1& c_1\\ a_2& d_2& c_2\\ a_3& d_3& c_3\end{array}\right|D_z=\left|\begin{array}{ccc}{a}_1& b_1& d_1\\ a_2& b_2& d_2\\ a_3& b_3& d_3\end{array}\right|$

Using Cramer’s Rule: $x=\frac{D_x}{D},y=\frac{D_y}{D},z=\frac{D_z}{D}$

• For a unique solution: $D\ne 0$.
• For an infinite number of solutions: $D_x=D_y=D_z=0$.
  • If all cofactors of $D$ are zero, then there are no solutions.
• For no solution (inconsistent system): Not all of $D_x$, $D_y$, and $D_z$ are zero.

Cayley-Hamilton Theorem

Every matrix satisfies its characteristic equation.

The characteristic equation of a matrix $A$ is given by: $|A-xI|=0$ where $I$ is the identity matrix and $x$ is any variable.

For a $2\times 2$ matrix: $A=\left[\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\right]$ The characteristic equation can be written as: $x^2-x(a_1+b_2)-(a_1{b}_2-a_2{b}_1)=0$

Important: Every matrix will satisfy its own characteristic equation.

Important Questions

1. Calculate product of following matrix

   1. $A=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$ && $B=\left[\begin{array}{ccc}2& 3& 4\end{array}\right]$

   2. $A=\left[\begin{array}{cc}6& 9\\ 2& 3\end{array}\right]$ && $B=\left[\begin{array}{ccc}2& 6& 0\\ 6& 9& 8\end{array}\right]$

2. $A=\left[\begin{array}{cc}7& 2\\ -1& 4\end{array}\right]$ && $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ then prove that (A - 5I)(A - 6I) = O

3. $A=\left[\begin{array}{ccc}1& -2& 0\\ 2& 1& 3\\ 0& -2& 1\end{array}\right]$ && $B=\left[\begin{array}{ccc}7& 2& -6\\ -2& 1& -3\\ -4& 2& 5\end{array}\right]$ then prove that (A - 5I)(A - 6I) = O

   1. Find AB

   2. Find answers to the following equations using Matrices and Pre multiplication method

      x - 2y = 10, 2x + y + 3z = 8, -2y + z = 7

4. Solve the following; $\left|\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right|$

5. Given $A=\left|\begin{array}{ccc}1& 0& 1\\ 0& 1& 2\\ 0& 0& 4\end{array}\right|$ prove that |3A| = 27|A| in two ways and also calculate the value of |3A|.

6. Find the equation of the line joining the points (1,2) and (3,6) using determinants

7. If (x,y) is any point on the line joining (a,0), (0,b) then show that x/a + y/b = 1

8. Evaluate $A=\left|\begin{array}{ccc}1& x& yz\\ 1& y& zx\\ 1& z& xy\end{array}\right|$ using cofactors of the elements of the third column.

9. Given $A=\left|\begin{array}{ccc}2& 1& 5\\ 3& -2& -4\\ -3& 1& -2\end{array}\right|$ show that $A\ast (adjA)=(adjA)\ast A=|A|I$ where $A=\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|$

10. Show that a unit matrix is a non singular matrix

11. If A is a non-singular matrix of order 3 and |adj(A)| = |A|^k then find value of k.

12. If A is a square matrix of order 3 such that |adj(A)| = 64 then find |A|

13. If $A=\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]$ then find |adj(A)|

14. If $A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$ show that A² - 5A + 7I = 0 and also find A^-1 (Do it by two different methods)

15. If $A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$ show that A² - 4A - 5I = 0 and also find A^-1 (Do it by two different methods)

16. Solve the following system of equations by matrix method

    5x + 2y + z = 8, 2x - y + 3z = 4, 2x - 2y + 4z = 5

17. If $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& -3\\ -3& 2& -4\end{array}\right]$ Find A^-1 and also solve the following system of equations

    x + 2y - 3z = -4, 2x + 3y + 2z = 14, 3x - 3y - 4z = -15

18. Examine the consistancy of following system of equations

    3x - y - 2z = 2, 2y - z = -1, 3x - 5y = 3

19. If A is a square matrix of order 3 and |A| = -4 then |adj(A)| is equal to ?

20. If $A=\left[\begin{array}{cc}\alpha & \beta \\ \beta & \alpha \end{array}\right]$ and |A|³ = 125 then what is the value of $\alpha$

    Options - $\pm 3$, $\pm 2$, $\pm 5$, 0

21. Find X and Y if $X+Y=\left[\begin{array}{cc}7& 0\\ 2& 5\end{array}\right]$, $X-Y=\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$

22. If co-ordinates of the vertices of a equilateral triangle with sides of length ‘a’ are (x₁,y₁),(x₂,y₂),(x₃,y₃) then show that $A=\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=\frac{3a^2}{4}$

23. Solve the following set of linear equations using matrix method:

    x - y + 2z = 7, 3x + 4y -5z = -5, 2x - y + 3z = 12

24. Find A - B $A=\left[\begin{array}{cc}si{n}^{-1}(\pi x)& ta{n}^{-1}(\frac{x}{\pi })\\ si{n}^{-1}(\frac{x}{\pi })& co{t}^{-1}(\pi x)\end{array}\right]B=\left[\begin{array}{cc}-co{s}^{-1}(\pi x)& ta{n}^{-1}(\frac{x}{\pi })\\ si{n}^{-1}(\frac{x}{\pi })& -ta{n}^{-1}(\pi x)\end{array}\right]$

25. If for a square matrix A, A² - 3A + I = 0 and A^-1 = xA + yI then find the value of x + y

26. Find the number of 3x3 non-singular matrix, with four entries as 1 and all the other entries as 0

    Options - less than 4, 5, 6, at least 7

27. If w != 1 is the complex cube root of unity and matrix $H=\left[\begin{array}{cc}w& 0\\ 0& w\end{array}\right]$ then find the value of H⁷⁰

28. If A is an 3x3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T then find the value of BB^T