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Matrix

Matrix
A rectangular arrangement of numbers arranged in rows and columns enclosed with large brackets or round brackets is known as matrix. Usually, matrix is represented by a capital letter. It’s plural is matrices. E.g.
A = \(\begin{bmatrix}a&b\\c&d\\\end{bmatrix}\)
B = \(\left(\begin{matrix} 1&2&3\\1&4&9\\\end{matrix}\right)\)

Types of matrix

  1. Row matrix
    A matrix having only one row is called a row matrix. For examples A = [2 4 6], B = [3, 4] are row matrices.
  2. Column matrix
    A matrix having only one column is called a column matrix. For examples A = \(\begin{bmatrix} 5\\6 \\\end{bmatrix}\) and B = \(\begin{bmatrix}p\\q\\r\\\end{bmatrix}\)
  3. Zero or null matrix
    A matrix with all element zero is called a zero or null matrix. For examples \(\begin{bmatrix}0&0\\0&0\\\end{bmatrix}\) and \(\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\\\end{bmatrix}\) are null matrices of order 2 \(\times\) 2 and 3 \(\times\) 3 respectively.
  4. Square matrix
    A matrix in which the number of rows and columns are equal is called a square matrix. For e.g. A = \(\begin{bmatrix}2&4\\4&16\\\end{bmatrix}\) and B = \(\begin{bmatrix}1&5&8\\3&4&6\\9&4&2\\\end{bmatrix}\) are square matrices.
  5. Diagonal matrix
    A square matrix in which elements in the leading diagonal are non-zero and zero is called a diagonal matrix. For example C = \(\begin{bmatrix} 3&0\\0&5\\\end{bmatrix}\) and \(\begin{bmatrix} 4&0&0\\0&8&0\\0&0&16\\\end{bmatrix}\)
  6. Scalar matrix
    A digital matrix in which all the diagonal elements are equal, is called a scalar matrix. For examples A = \(\begin{bmatrix}4&0\\0&4\\\end{bmatrix}\) and B = \(\begin{bmatrix}x&0&0\\0&x&0\\0&0&x\\\end{bmatrix}\) are scalar matrices.
  7. Identity or unit matrix
    A square matrix in which the elements in the leading diagonal are 1 and rest are zero, is called a unit matrix or an identity matrix. It is denoted by I. For examples I = \(\begin{bmatrix} 1&0\\0&1\\\end{bmatrix}\) and I = \(\begin{bmatrix} 1&0&0\\0&1&0\\ 0&0&1\\\end{bmatrix}\) are unit matrices of order 2 \(\times\) 2 and 3 \(\times\) 3 respectively.
  8. Equal matrix
    Two matrices are said to be equal, if they are of the same order and their corresponding elements are equal. For examples if A = \(\begin{bmatrix}2&4\\6&8\\\end{bmatrix}\) and B = \(\begin{bmatrix}2&4\\6&8\\\end{bmatrix}\) then, A = B.
    Thus, \(\begin{bmatrix} p&q\\r&s\\\end{bmatrix}\) = \(\begin{bmatrix}6&8\\3&6\\\end{bmatrix}\) if and only if, p = 6, q = 8, r = 3 and s = 6.

Examples

  • Construct a 2 \(\times\) 2 matrix whose elements are given by aij = \(\frac{i}{2j}\)
    Solution:
    Let a 2 \(\times\) 2 matrix is \(\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{bmatrix}\)
    Now,
    a11 = \(\frac{1}{2\times1}\) = \(\frac{1}{2}\)
    a12 = \(\frac{1}{2\times2}\) = \(\frac{1}{4}\)
    a21 = \(\frac{2}{2\times1}\) = \(\frac{2}{2}\) = 1
    a22 = \(\frac{2}{2\times2}\) = \(\frac{2}{4}\) = \(\frac{1}{2}\)
    \(\therefore\) The required matrix is \(\begin{bmatrix} \frac{1}{2}& \frac{1}{4}\\1&\frac{1}{2}\\\end{bmatrix}\)
  • Find the value of x, y, if \(\left(\begin{matrix} x + y \\ x – y \\\end{matrix}\right)\) = \(\left(\begin{matrix} 5\\3\\\end{matrix}\right)\).
    Solution:
    x + y = 5 -------(i)
    x – y = 3 -------(ii)
    Adding equation (i) and (ii)
    \(\begin{matrix}
    &x& +& y& =& 5&\\
    &x& -& y& =& 3&\\
    (+)&&&&&&\\
    \hline
    &2x&&&=&8\\
    \end{matrix}\)
    or, x = \(\frac{8}{2}\)
    \(\therefore\) x = 4
    Putting the value of x = 4 in equation (i)
    x + y = 5
    or, 4 + y = 5
    or, y = 5 – 4
    \(\therefore\) y = 1
    \(\therefore\) x = 4 and y = 1

Transpose of a matrix
Let A = \(\begin{bmatrix}2&3&5\\6&7&0\\\end{bmatrix}\) be a 2 \(\times\) 3 matrix. Interchanging the rows and columns of A we get a new matrix of order 3 \(\times\) 2. The new matrix is \(\begin{bmatrix}2&6\\3&7\\5&0\\\end{bmatrix}\) and is denoted by A’ or AT.
\(\therefore\) AT = \(\begin{bmatrix}2&6\\3&7\\5&0\\\end{bmatrix}\). Thus, for a given matrix A, a new matrix formed by interchanging its rows and columns is called the transpose of A.

Properties of Transpose

  • (AT)T = A
  • (A + B)T = AT + BT
  • (KA)T = KAT

Examples

  • If A = \(\begin{bmatrix}1&3\\2&4\\\end{bmatrix}\), B = \(\begin{bmatrix}0&3\\2&1\\\end{bmatrix}\) and C = \(\begin{bmatrix}5&7\\6&8\\\end{bmatrix}\). Find A – B + C.
    Solution:
    A – B + C = \(\begin{bmatrix}1&3\\2&4\\\end{bmatrix}\) - \(\begin{bmatrix}0&3\\2&1\\\end{bmatrix}\) + \(\begin{bmatrix}5&7\\6&8\\\end{bmatrix}\)
    = \(\begin{bmatrix}1-0+5&3-3+7\\2-2+6&4-1+8\\\end{bmatrix}\)
    = \(\begin{bmatrix}6&7\\6&11\\\end{bmatrix}\)
  • If A = \(\begin{bmatrix}1&2\\3&4\\\end{bmatrix}\), B = \(\begin{bmatrix}1&0\\2&3\\ \end{bmatrix}\), then prove the following A + B = B + A
    Solution:
    L.H.S. = A + B = \(\begin{bmatrix}1&2\\3&4\\\end{bmatrix}\) + \(\begin{bmatrix}1&0\\2&3\\\end{bmatrix}\)
    = \(\begin{bmatrix}1+1&2+0\\3+2&4+3\\\end{bmatrix}\)
    = \(\begin{bmatrix}2&2\\5&7\\\end{bmatrix}\)
    R.H.S. = B + A = \(\begin{bmatrix}1&0\\2&3\\\end{bmatrix}\) + \(\begin{bmatrix}1&2\\3&4\\\end{bmatrix}\)
    = \(\begin{bmatrix}1+1&0+2\\2+3&3+4\\\end{bmatrix}\)
    = \(\begin{bmatrix}2&2\\5&7\\\end{bmatrix}\)
    \(\therefore\) L.H.S. = R.H.S.

Multiplication of Matrix

Examples

  • If A = \(\left(\begin{matrix}1&-1\\-3&3\\\end{matrix}\right)\) and B = \(\left(\begin{matrix}2&-5\\2&-5\\\end{matrix}\right)\). Show that AB is a null matrix.
    Solution:
    AB = \(\left(\begin{matrix}1&-1\\-3&3\\\end{matrix}\right)\)\(\left(\begin{matrix}2&-5\\2&-5\\\end{matrix}\right)\)
    = \(\left(\begin{matrix}1\times 2 + (-1)\times2& 1\times(-5) + (-1)\times(-5)\\ -3\times2 + 3\times2&-3\times(-5) + 3\times(-5)\\\end{matrix}\right)\)
    = \(\left(\begin{matrix}2-2&-5+5\\-6+6&15-15\\\end{matrix}\right)\)
    = \(\left(\begin{matrix}0&0\\0&0\\\end{matrix}\right)\)
    \(\therefore\) AB is a null matrix.
  • If A = \(\left(\begin{matrix}3&4\\2&3\\\end{matrix}\right)\) and B = \(\left(\begin{matrix} 3&-4\\-2&3\\\end{matrix}\right)\) then, Prove that AB is an identity matrix.
    Solution:
    AB = \(\left(\begin{matrix}3&4\\2&3\\\end{matrix}\right)\)\(\left(\begin{matrix}3&-4\\-2&3\\\end{matrix}\right)\)
    = \(\left(\begin{matrix}3\times3 + 4\times(-2)&3\times(-4) + 4\times3\\2\times3 + 3\times(-2)&2\times(-4) + 3\times3\\\end{matrix}\right)\)
    = \(\left(\begin{matrix}9-8&-12+12\\6-6&-8+9\\\end{matrix}\right)\)
    = \(\left(\begin{matrix}1&0\\0&1\\\end{matrix}\right)\)
    \(\therefore\) The matrix AB is a identity matrix.

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