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DETERMINANTS

A determinant is a number associated to a square matrix. Determinants are possible only for square matrices. DETERMINANTS. TYPES OF DETERMINANTS. DETERMINANT OF ORDER ONE- let A=[a 11 ] be a matrix of order 1,then its determinant is defined as : |A|=|a 11 |=a 11 .

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DETERMINANTS

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  1. A determinant is a number associated to a square matrix. Determinants are possible only for square matrices. DETERMINANTS

  2. TYPES OF DETERMINANTS • DETERMINANT OF ORDER ONE- let A=[a11] be a matrix of order 1,then its determinant is defined as : |A|=|a11|=a11. • DETERMINANT OF ORDER 2-the value is obtained by multiplying the two elements on the principal diagonal and subtracting the product of elements on the cross diagonal. • DETERMINANT OF ORDER 3-the value is obtained by multiplying each element of first row by a determinant obtained by deleting from the given determinant ,the row and the column to which element belongs ,the sign being taken positive and negative alternatively.

  3. MINORS AND COFACTORS • MINORS- let A be a square matrix of order 3 ,then minor of an element a11 is the determinant of the sub matrix obtained from A by deleting first row and first column containing the elements a11.It is donated by M11. • COFACTORS- The minor of an element with a proper sign is called the cofactor to the element.

  4. DIFFERENCE BETWEEN DETERMINANT $ MATRIX • A matrix may be rectangular or a square but the determinant is always square. • A matrix has no definite value but a determinant has a definite value. • We use square brackets [ ] to denote a matrix but we use two vertical lines | | to denote for determinant.

  5. PROPERTIES OF DETERMINANTS • If the rows and columns of determinants are interchanged ,the value of the determinant remains unchanged i.e. |A|=|A’|. • If any two adjacent rows or columns of a determinant are interchanged, the value of determinant changes only in sign. • If any two rows or columns of a determinant are identical or are multiple of each other, then the value of determinant is zero.

  6. If all the elements of any row or column of a determinant are zero, then the value of determinant is zero. • If all the elements of any row or column of a determinant are multiplied by a quantity (K), the value of the determinant is multiplied by same quantity. • If each element of any row or column of a determinant is sum of two elements ,the determinant can be expressed as the sum of two determinants of the same order.

  7. The addition of a constant multiple of one row or column to another row or column leave the determinant unchanged. • The determinant of the product of the two matrices of same order is equal to the product of individual determinants.

  8. SYMBOLS & TERMINOLOGIES • R1, R2 ,R3 , etc, stands for first row, second row, third row, etc. • C1, C2 ,C3 etc, stands for first column, second column, third column, etc. • Operate R2 = R2 – R1 means ‘from the element of the second row subtract the corresponding elements of the first row’. • Operate C2 = C1 + 2C3 means ‘two the elements of the first column, add twice the corresponding elements of the third column.

  9. THANK YOU

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