CBSE Class 12 Mathematics Matrices and Determinants Assignment Set 03

Key Notes

• A matrix is an ordered rectangular array of numbers or functions.
• A matrix having \( m \) rows and \( n \) columns is called a matrix of order \( m \times n \).
• \( [a_{ij}]_{m \times 1} \) is a column matrix.
• \( [a_{ij}]_{1 \times n} \) is a row matrix.
• An \( m \times n \) matrix is a square matrix if \( m = n \).
• \( A = [a_{ij}]_{m \times n} \) is a diagonal matrix if \( a_{ij} = 0 \), when \( i \neq j \).
• \( A = [a_{ij}]_{n \times n} \) is a scalar matrix if \( a_{ij} = 0 \) when \( i \neq j \), and \( a_{ij} = k \) (\( k \) is some constant), when \( i = j \).
• \( A = [a_{ij}]_{n \times n} \) is an identity matrix, if \( a_{ij} = 1 \), when \( i = j \), and \( a_{ij} = 0 \), when \( i \neq j \).
• A zero matrix has all its elements as zero.
• \( A = [a_{ij}] = [b_{ij}] = B \) if (i) \( A \) and \( B \) are of same order, (ii) for all possible values of \( i \) and \( j \).
• \( kA = k[a_{ij}]_{m \times n} = [k(a_{ij})]_{m \times n} \)
• \( -A = (-1)A \)
• \( A - B = A + (-1)B \)
• \( A + B = B + A \)
• \( (A + B) + C = A + (B + C) \), where \( A \), \( B \) and \( C \) are of same order.
• \( k(A + B) = kA + kB \), where \( A \) and \( B \) are of same order, \( k \) is constant.
• \( (k + l)A = kA + lA \), where \( k \) and \( l \) are constant.
• If \( A = [a_{ij}]_{m \times n} \) and \( B = [b_{jk}]_{n \times p} \), then \( AB = C = [c_{ik}]_{m \times p} \), where \( c_{il} = \sum_{j=1}^{n} a_{ij}b_{jl} \)

(i) \( A(BC) = (AB)C \)

(ii) \( A(B + C) = AB + AC \)

(iii) \( (A + B)C = AC + BC \)

• If \( A = [a_{ij}]_{m \times n} \), then \( A' \) or \( A^T = [a_{ji}]_{n \times m} \)
• (i) \( (A')' = A \)
• (ii) \( (kA)' = kA' \)
• (iii) \( (A + B)' = A' + B' \)
• (iv) \( (AB)' = B'A' \)
• \( A \) is a symmetric matrix if \( A' = A \).
• \( A \) is a skew symmetric matrix if \( A' = -A \).
• Any square matrix can be represented as the sum of a symmetric and a skew symmetric matrix.
• Elementary operations of a matrix are as follows:
  • (i) \( R_i \leftrightarrow R_j \) or \( C_i \leftrightarrow C_j \)
  • (ii) \( R_i \rightarrow kR_i \) or \( C_i \rightarrow kC_i \)
  • (iii) \( R_i \rightarrow R_i + kR_j \) or \( C_i \rightarrow C_i + kC_j \)
• If \( A \) and \( B \) are two square matrices such that \( AB = BA = I \), then \( B \) is the inverse matrix of \( A \) and is denoted by \( A^{-1} \), and \( A \) is the inverse of \( B \).
• Inverse of a square matrix, if it exists, is unique.