Advance Math Homework Help
We help you solve Advance Math homework, assignments and projects with step by step explanation, formulas and concepts. Our step by step explanation helps you to understand the solution and proofs in a better ways.
How We Explain the Steps of a Advance Math Homework Questions
Example 1: What is a Matrix ? Explain addition, subtraction and multiplication of two matrices.
Answer :
Matrix : A matrix is an arrangement of elements into rows and columns.
Examples
- \(\left(\begin{array}{ccc}3 & -2 & 1 \\4 & -5 & 8\\\end{array}\right)\) is a matrix with two rows and three columns.
- \(\left(\begin{array}{ccc}2 & 3 & 4 \\3 & 2 & 1 \\4 & 6 & 8 \\\end{array}\right)\) is a matrix with three rows and three columns
Addition and Subtraction of Two Matrices : If Two matrices have equal number of rows and columns are equal then they can be added element wise.
Let \(A=\left(\begin{array}{ccc}0 & -1 & 2 \\-3 & 2 & 0 \\4 & 2 & 0 \\\end{array}\right) and B=\left(\begin{array}{ccc}9 & 6 & 3 \\-5 & 0 & 2 \\3 & 3 & 2 \\\end{array}\right)\)
Both A and B have equal number of rows and matrices. So we can find their addition and subtraction as follows
Addition of A and B
\[\left(\begin{array}{ccc}0+9 & -1+6 & 2+3 \\-3-5 & 2+0& 0+2\\ 4+3&2+3&0+2\\\end{array}\right)=\left(\begin{array}{ccc}9 & 5 & 5 \\-8 & 2& 2\\ 7&5&2\\\end{array}\right) \]
Subtraction of A and B
\[\left(\begin{array}{ccc}0-9 & -1-6 & 2-3 \\-3-(-5) & 2-0& 0-2\\ 4-3&2-3&0-2\\\end{array}\right)=\left(\begin{array}{ccc}-9 & -7 & -1 \\2 & 2& -2\\ 1&-1&-2\\\end{array}\right) \]
Scalar Multiplication : When we want to multiply by a number \(k\) to a matrix then it is multiplied with each entry \( a(i, j) \) of matrix \(A\) and it is called scalar multiplication of k with A.
Example : Suppose \(k = 5\) and \(A=\left(\begin{array}{ccc}2 & -1 & 1 \\0 & 4 & 2 \\0 & -3 & -5 \\\end{array}\right)\) then scalar multiplication of k with A is
\[kA=5A = \left(\begin{array}{ccc}2*5 & -1*5 & 1*5 \\0*5 & 4*5 & 2*5 \\0*5 & -3*5 & -5*5 \\\end{array}\right) = \left(\begin{array}{ccc}10 & -5 & 5 \\0 & 20 & 10\\0 & -15 & -25 \\\end{array}\right)\]
Multiplication of Two Matrices :
If number of rows of a matrix A are equal to number of columns of a matrix B then the product AB is find as follows
Let \(A =\left(\begin{array}{cc}2 & 8 \\3 & 4 \end{array}\right), B =\left(\begin{array}{cc}-3 & 1 \\0 & -1 \end{array}\right)\)
Clearly number of rows in A are two equal to number of columns of B equal to 2.
\[AB =\left(\begin{array}{cc}2 & 8 \\3 & 4 \end{array}\right)\left(\begin{array}{cc}-3 & 1 \\0 & -1 \end{array}\right)\]
\[=\left(\begin{array}{cc}2*(-3)+8*0 & 2*1+8*(-1) \\3*(-3)+4*0 & 3*1+4*(-1) \end{array}\right) \]
\[AB=\left(\begin{array}{cc}-6 & -6 \\-9 & -1 \end{array}\right)\]
