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

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.

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.

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.

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.

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.

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.

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.

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