Lecture Two: Determinants, Inverse Matrices and Systems of Equations

Special Types of Matrices

Symmetric Matrices

• Definition: A matrix $A$ is symmetric if $A^T=A$. In other words, the transpose of the matrix is equal to the matrix itself.

• Properties:

  • A symmetric matrix must always be square (i.e., the number of rows must equal the number of columns).
  • The elements across the main diagonal of a symmetric matrix are symmetric with respect to the diagonal.

• Example:

$$A=\left[\begin{array}{ccc}1& x& -y\\ x& 2& z\\ -y& z& 3\end{array}\right]$$

In this case, the matrix is symmetric because $A^T=A$.

• Example 1: Show that the matrix $A$ is symmetric:$$A=\left[\begin{array}{ccc}1& 2& 4\\ 2& 0& 17\\ 4& 17& 6\end{array}\right]$$$$A^T=\left[\begin{array}{ccc}1& 2& 4\\ 2& 0& 17\\ 4& 17& 6\end{array}\right]$$$$A=A^T$$Hence, $A$ is a symmetric matrix.

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Skew-Symmetric Matrices

• Definition: A matrix $A$ is skew-symmetric if $A^T=-A$. In other words, the transpose of the matrix is the negative of the matrix.

• Properties:

  • A skew-symmetric matrix must be square (i.e., the number of rows must equal the number of columns).
  • All diagonal entries of a skew-symmetric matrix must be 0, because the transpose of the diagonal element must equal its negative, which is only possible if it is zero.

• Example:

  $$A=\left[\begin{array}{ccc}0& x& y\\ -x& 0& -z\\ -y& z& 0\end{array}\right]$$

  This matrix is skew-symmetric because $A^T=-A$.

Determinants

Determinant of a $2\times 2$ Matrix

For a matrix $A$ of size $2\times 2$:

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$

The determinant is calculated as:

$$det(A)=ad-bc$$

where $ad-bc\ne 0$.

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Determinant of a $3\times 3$ Matrix

For a matrix $A$ of size $3\times 3$:

$$A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$$

The determinant is calculated as:

$$|A|=a\left|\begin{array}{cc}e& f\\ h& i\end{array}\right|-b\left|\begin{array}{cc}d& f\\ g& i\end{array}\right|+c\left|\begin{array}{cc}d& e\\ g& h\end{array}\right|$$

This simplifies to:

$$|A|=a(ei-fh)-b(di-fg)+c(dh-eg)$$

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Sign Rule for $3\times 3$ Determinants

To calculate the determinant of a $3\times 3$ matrix, you can use the sign rule based on the positions of the elements in the matrix. The signs alternate in a checkerboard pattern, as shown below:

$$\left|\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right|$$

This pattern helps determine the signs when calculating the determinant by cofactor expansion along any row or column.

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Examples

• Example 1: Determinant of a $2\times 2$ Matrix:

$$B=\left[\begin{array}{cc}1& 4\\ 2& 6\end{array}\right]$$

The determinant of $B$ is:

$$|B|=1(6)-4(2)=-2$$

• Example 2: Determinant of a $3\times 3$ Matrix:

$$F=\left[\begin{array}{ccc}1& -2& 7\\ 6& 0& 1\\ 3& 10& 4\end{array}\right]$$

Using cofactor expansion, we get:

$$|F|=1\left|\begin{array}{cc}0& 1\\ 10& 4\end{array}\right|-(-2)\left|\begin{array}{cc}6& 1\\ 3& 4\end{array}\right|+7\left|\begin{array}{cc}6& 0\\ 3& 10\end{array}\right|$$

Calculating the smaller determinants:

$$|F|=1(0-10)+2(24-3)+7(60-0)$$$$|F|=-10+42+420=452$$

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Properties of Determinants

For any square matrix $A$ of size $n\times n$ and scalar $k\in \mathbb{R}$, the following properties hold:

• Multiplication Property:

$$det(AB)=det(A)\cdot det(B)$$

• Scalar Multiplication Property:

$$det(kA)=k^n\cdot det(A)$$

• Transpose Property:$$det(A^T)=det(A)$$

> $\det(A)$ is often written as $|A|$.

Inverse of a Matrix

Definition

An $n\times n$ matrix $A$ has an inverse if its determinant is non-zero ($det(A)\ne 0$), and the inverse $A^{-1}$ satisfies the following condition:

$$A\cdot A^{-1}=A^{-1}\cdot A=I_n$$

Where $I_n$ is the identity matrix of size $n\times n$. If matrix $A$ has an inverse, it is called non-singular (or invertible). If it does not have an inverse, it is called singular (or non-invertible).

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Theorem: Uniqueness of the Inverse

A matrix can have only one inverse.

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Adjoint of a Matrix

The adjoint of a square $2\times 2$ matrix $A$ is defined as:

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],\text{adj}(A)=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$$

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Theorem: Singularity

Any matrix is singular (non-invertible) if $det(A)=0$. Conversely, a matrix is non-singular (invertible) if $det(A)\ne 0$.

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Properties of Inverse Matrices

For square matrices $A$ and $B$, and scalar $k\in \mathbb{R}$, the following properties hold:

• Product Property:

$$(AB{)}^{-1}=B^{-1}{A}^{-1}$$

• Transpose Property:

$$(A^{-1}{)}^T=(A^T{)}^{-1}$$

• Determinant Property:

$$det(A^{-1})=\frac{1}{det(A)}$$

• Scalar Multiplication Property:$$(kA{)}^{-1}=\frac{1}{k}{A}^{-1}$$

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Examples

• Example 1: Show that $A$ is non-singular.

  Given matrix $A$:

$$A=\left[\begin{array}{ccc}1& 0& 4\\ -2& 0& 17\\ 4& 17& 6\end{array}\right]$$

The determinant of $A$ is calculated as:

$$|A|=1\left|\begin{array}{cc}0& 17\\ 17& 6\end{array}\right|-0\left|\begin{array}{cc}-2& 17\\ 4& 6\end{array}\right|+4\left|\begin{array}{cc}-2& 0\\ 4& 17\end{array}\right|$$

Simplifying:

$$|A|=1(-289)-0+4(-34)=-425$$

Since $|A|\ne 0$, matrix $A$ is non-singular.

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• Example 2: Find the values of $k$ that make matrix $A$ singular.

  Given matrix $A$:

$$A=\left[\begin{array}{cc}k& 2\\ -2& k+5\end{array}\right]$$

For singularity, we require the determinant to be 0:

$$|A|=k(k+5)-2(-2)=0$$

Simplifying:

$$k^2+5k+4=0$$

Factoring the quadratic equation:

$$(k+1)(k+4)=0$$

Hence, $k=-1$ or $k=-4$. Therefore, the matrix $A$ is singular when $k=-1$ or $k=-4$.

Inverse of a $2\times 2$ Matrix

The inverse of a $2\times 2$ matrix $A$:

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$

If matrix $A$ is a $2\times 2$ matrix and has a non-zero determinant, its inverse is given by:

$$A^{-1}=\frac{1}{det(A)}\cdot \text{adj}(A)$$

Which simplifies to:

$$A^{-1}=\frac{1}{ad-bc}\cdot \left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$$

Inverse of a $3\times 3$ Matrix

For a $3\times 3$ matrix $A$, the inverse exists if and only if the determinant of $A$ is non-zero. The inverse is calculated as:

$$A^{-1}=\frac{1}{det(A)}\cdot \text{adj}(A)$$

Where:

• $det(A)$ is the determinant of $A$.
• $\text{adj}(A)$ is the adjoint of $A$, which is the transpose of its cofactor matrix.

Matrix Representation

If $A$ is given as:

$$A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$$

1. Determinant ($det(A)$)
   The determinant of $A$ is calculated as:

   $$det(A)=a(ei-fh)-b(di-fg)+c(dh-eg)$$

2. Cofactor Matrix
   The cofactor matrix is obtained by calculating the determinant of the minor for each element of $A$ following the Sign rule.

The cofactor matrix for $A$ is:

$$\text{Cofactor}(A)=\left[\begin{array}{ccc}e(i)-f(h)& -d(i)-f(g)& d(h)-e(g)\\ -b(i)-c(h)& a(i)-c(g)& -a(h)-b(g)\\ b(f)-c(e)& -a(f)-c(d)& a(e)-b(d)\end{array}\right]$$

3. Adjugate Matrix
   The adjugate of $A$ is the transpose of the cofactor matrix:

   $$\text{adj}(A)=\text{Cofactor}(A{)}^T$$

4. Inverse Formula
   Using the determinant and adjugate, the inverse is given by:

   $$A^{-1}=\frac{1}{det(A)}\cdot \text{adj}(A)$$

Simplified Example

For a matrix:

$$A=\left[\begin{array}{ccc}2& 1& 3\\ 1& 0& 2\\ 3& 1& 4\end{array}\right]$$

• Calculate $|A|$ using the formula.

  $$|A|=a(ei-fh)-b(di-fg)+c(dh-eg)$$$$|A|=2(0-2)-1(4-6)+3(1-0)=1$$

• Find the adjoint.

  $$adj(A)=\left[\begin{array}{ccc}0(4)-1(2)& 1(4)-2(3)& 1(1)-0(3)\\ 1(4)-1(3)& 2(4)-3(3)& 2(1)-1(3)\\ 1(2)-0(3)& 2(2)-1(3)& 2(0)-1(1)\end{array}\right]$$$$adj(A)=\left[\begin{array}{ccc}-2& -2& 1\\ 1& -1& -1\\ 2& 1& -1\end{array}\right]$$

  Applying the Sign rule:

  $$adj(A)=\left[\begin{array}{ccc}-2& 2& 1\\ -1& -1& 1\\ 2& -1& -1\end{array}\right]$$

• Transpose the adjoint.

  $${\left[\begin{array}{ccc}-2& 2& 1\\ -1& -1& 1\\ 2& -1& -1\end{array}\right]}^T=\left[\begin{array}{ccc}-2& -1& 2\\ 2& -1& -1\\ 1& 1& -1\end{array}\right]$$

• Substitute into the inverse formula.

  $$A^{-1}=\frac{1}{det(A)}\cdot \text{adj}(A)$$$$A^{-1}=\frac{1}{1}\cdot \left[\begin{array}{ccc}-2& -1& 2\\ 2& -1& -1\\ 1& 1& -1\end{array}\right]$$$$A^{-1}=\left[\begin{array}{ccc}-2& -1& 2\\ 2& -1& -1\\ 1& 1& -1\end{array}\right]$$

Inverse and Systems of Equations

System of Equations and Matrix Representation

A system of linear equations can be represented in matrix form as:

$$A\cdot X=B$$

Where:

• $A$ is the coefficient matrix,
• $X$ is the column vector of unknowns, and
• $B$ is the column vector of constants (the right-hand side of the equations).

If $A^{-1}$ (the inverse of matrix $A$) exists, the solution to the system of equations is given by:

$$X=A^{-1}\cdot B$$

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Theorem: Solution Existence

The $n\times n$ system of equations $A\cdot X=B$ has a solution if $A^{-1}$ exists. The solution is then obtained by calculating:

$$X=A^{-1}\cdot B$$

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Example 6: Condition for a Solution

Consider the system of equations:

$$x+y=1$$$$2x+ky=0$$

The coefficient matrix $A$ is:

$$A=\left[\begin{array}{cc}1& 1\\ 2& k\end{array}\right]$$

The determinant of $A$ is:

$$|A|=1(k)-2(1)=k-2$$

For the system to have a solution, $|A|\ne 0$. Setting the determinant equal to zero:

$$k-2=0\Rightarrow k=2$$

Thus, the system has a solution when $k\ne 2$.

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Example 7: Solving a System Using the Inverse of a 2x2 Matrix

Solve the following system of equations for $x$ and $y$:

$$2x+y=8$$$$-2x+3y=16$$

The coefficient matrix $A$, unknown vector $X$, and right-hand side vector $B$ are:

$$A=\left[\begin{array}{cc}2& 1\\ -2& 3\end{array}\right],X=\left[\begin{array}{c}x\\ y\end{array}\right],B=\left[\begin{array}{c}8\\ 16\end{array}\right]$$

To solve for $X=A^{-1}\cdot B$, first find the inverse of $A$:

$$A^{-1}=\frac{1}{2(3)-1(-2)}\cdot \left[\begin{array}{cc}3& -1\\ 2& 2\end{array}\right]$$$$A^{-1}=\frac{1}{8}\cdot \left[\begin{array}{cc}3& -1\\ 2& 2\end{array}\right]$$

Now, calculate $X=A^{-1}\cdot B$:

$$\left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{8}\cdot \left[\begin{array}{cc}3& -1\\ 2& 2\end{array}\right]\cdot \left[\begin{array}{c}8\\ 16\end{array}\right]$$$$=\frac{1}{8}\cdot \left[\begin{array}{c}3(8)-1(16)\\ 2(8)+2(16)\end{array}\right]$$$$\frac{1}{8}\cdot \left[\begin{array}{c}8\\ 48\end{array}\right]=\left[\begin{array}{c}1\\ 6\end{array}\right]$$$$\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}1\\ 6\end{array}\right]$$

Thus, $x=1$ and $y=6$.

Cramer's Rule for Solving Systems of Equations

Cramer's Rule provides a method to solve a system of linear equations using determinants. For a system represented by $A\cdot X=B$, where $A$ is an $n\times n$ matrix, the solution is given by the following steps.

Definitions:

• $\mathrm{\Delta }$: The determinant of the coefficient matrix $A$.
• ${\mathrm{\Delta }}_x$: The determinant of the matrix obtained by replacing the first column of $A$ with $B$.
• ${\mathrm{\Delta }}_y$: The determinant of the matrix obtained by replacing the second column of $A$ with $B$.

Solution for 2 Variables

For a system with two variables $x$ and $y$, the solution is:

$$x=\frac{{\mathrm{\Delta }}_x}{\mathrm{\Delta }},y=\frac{{\mathrm{\Delta }}_y}{\mathrm{\Delta }}$$

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Example 8: Solving a System Using Cramer's Rule

Solve the following system for $x$ and $y$ using Cramer's Rule:

$$2x+y=8$$$$-2x+3y=16$$

First, calculate the determinant of the coefficient matrix $A$:

$$\mathrm{\Delta }=\left|\begin{array}{cc}2& 1\\ -2& 3\end{array}\right|=2(3)-(-2)(1)=6+2=8$$

Next, calculate ${\mathrm{\Delta }}_x$ by replacing the first column of $A$ with $B$:

$${\mathrm{\Delta }}_x=\left|\begin{array}{cc}8& 1\\ 16& 3\end{array}\right|=8(3)-16(1)=24-16=8$$

Now, calculate ${\mathrm{\Delta }}_y$ by replacing the second column of $A$ with $B$:

$${\mathrm{\Delta }}_y=\left|\begin{array}{cc}2& 8\\ -2& 16\end{array}\right|=2(16)-(-2)(8)=32+16=48$$

Finally, solve for $x$ and $y$:

$$x=\frac{{\mathrm{\Delta }}_x}{\mathrm{\Delta }}=\frac{8}{8}=1$$$$y=\frac{{\mathrm{\Delta }}_y}{\mathrm{\Delta }}=\frac{48}{8}=6$$

Thus, the solution is $x=1$ and $y=6$.

Key Takeaways

• Symmetric matrices are equal to their transposes and must be square, with the elements mirrored across the main diagonal.
• Skew-symmetric matrices are the negative of their transposes and must be square, with all diagonal elements equal to zero.
• The determinant of a $2\times 2$ matrix is calculated as $ad-bc$.
• For a $3\times 3$ matrix, the determinant can be computed using cofactor expansion along any row or column.
• Determinants have several important properties, such as being invariant under transpose and related to the matrix multiplication.
• A matrix is invertible (non-singular) if its determinant is non-zero.
• The inverse of a $2\times 2$ matrix can be found using the adjoint and determinant.
• The inverse matrix properties allow for operations such as multiplication and transposition of matrices.
• A system of equations can be solved using the inverse of the coefficient matrix if it is invertible.
• Cramer's Rule provides an alternative method for solving systems using determinants.
• Both methods give the same results, and the system has a solution if the determinant of the coefficient matrix is non-zero.