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A determinant of order n is a square array of elements having n rows and n columns, there thus being n elements in each row and each column Now let aij denote the element at the intersection of the i th row and j th column If we then strike out the row and column in which aij appears, the determinant that remains is of order n 1 and is called the minor determinant of element aij In Chap 3 we showed that the value of a determinant is equal to the sum of the products of the elements of any row or column and their corresponding minor determinants, each such product being multiplied by 1 i j .

Thus, if we expand the thirdorder determinant in the last expression for x1 above, using minors of the rst column, we have that a22 a23 y2 a12 a13 y3 a12 a13 1 x 1 y1 a a a32 a33 a23 D 32 a33 22 Likewise, if we expand the third-order determinant in the last expression for x2 , in terms of the minors of the second column, we have that a21 a23 y2 a11 a13 y3 a11 a13 1 x2 y1 a a a a33 a23 D 31 31 a33 21 and lastly, if we expand the third-order determinant in the last expression for x3 , in terms of the minors of the third column, we have that a21 a22 y2 a11 a12 y3 a11 a12 1 x 3 y1 a a a a32 a22 D 31 31 a32 21 Now, in the last three equations above, let us denote the value of each second-order minor determinant, including the sign factor 1 i j , by the notation Aij , where i and j are the numbers of the row and column struck out to form the minor determinant.

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constraints.anchor = GridBagConstraints.EAST; constraints.insets = new Insets(5, 5, 5, 0); layout.setConstraints(passwordLabel, constraints); settingsPanel.add(passwordLabel); passwordField = new JPasswordField(); constraints = new GridBagConstraints(); constraints.anchor = GridBagConstraints.WEST; constraints.fill = GridBagConstraints.HORIZONTAL; constraints.gridwidth = GridBagConstraints.REMAINDER; constraints.insets = new Insets(5, 5, 5, 5); constraints.weightx = 1.0D; layout.setConstraints(passwordField, constraints); settingsPanel.add(passwordField); JLabel smtpServerLabel = new JLabel("SMTP Server:"); constraints = new GridBagConstraints(); constraints.anchor = GridBagConstraints.EAST; constraints.insets = new Insets(5, 5, 5, 0); layout.setConstraints(smtpServerLabel, constraints); settingsPanel.add(smtpServerLabel); smtpServerTextField = new JTextField(25); constraints = new GridBagConstraints(); constraints.gridwidth = GridBagConstraints.REMAINDER; constraints.insets = new Insets(5, 5, 5, 5); constraints.weightx = 1.0D; layout.setConstraints(smtpServerTextField, constraints); settingsPanel.add(smtpServerTextField); // Set up buttons panel. JPanel buttonsPanel = new JPanel(); JButton connectButton = new JButton("Connect"); connectButton.addActionListener(new ActionListener() { public void actionPerformed(ActionEvent e) { actionConnect(); } }); buttonsPanel.add(connectButton); JButton cancelButton = new JButton("Cancel"); cancelButton.addActionListener(new ActionListener() { public void actionPerformed(ActionEvent e) { actionCancel(); } }); buttonsPanel.add(cancelButton); // Add panels to display.

* Using this notation, the last three equations above become A11 A A y 21 y2 31 y3 D 1 D D A A A x2 12 y1 22 y2 32 y3 D D D A A A x3 13 y1 23 y2 33 y3 D D D x1 which in matrix notation becomes 2 3 2 A11 x1 6 7 16 x2 5 4 A12 4 D x3 A13 32 3 479.

Equation (479) above is the inverse form of eqs. (476) and (477); that is, (479) is of the form X A 1 Y 480

getContentPane().setLayout(new BorderLayout()); getContentPane().add(settingsPanel, BorderLayout.CENTER); getContentPane().add(buttonsPanel, BorderLayout.SOUTH); // Size dialog box to components. pack(); // Center dialog box over application. setLocationRelativeTo(parent); } // Validate connection settings and close dialog box. private void actionConnect() { if (serverTextField.getText().trim().length() < 1 || usernameTextField.getText().trim().length() < 1 || passwordField.getPassword().length < 1 || smtpServerTextField.getText().trim().length() < 1) { JOptionPane.showMessageDialog(this, "One or more settings is missing.", "Missing Setting(s)", JOptionPane.ERROR_MESSAGE); return; } // Close dialog box. dispose(); } // Cancel connecting and exit program. private void actionCancel() { System.exit(0); } // Get e-mail server type. public String getType() { return (String) typeComboBox.getSelectedItem(); } // Get e-mail server. public String getServer() { return serverTextField.getText(); } // Get e-mail username. public String getUsername() { return usernameTextField.getText();

* Aij is called the cofactor of element aij . If Mij is the minor determinant of any element aij , the cofactor of aij is the minor determinant with proper sign included; thus Aij 1 i j Mij

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